Usability

Usability
Usability

D9d

Proposal for exemplars and work towards them:

Usability

Workpackage 9 Deliverable

Date: 30th November 2005

IST project contract no. 507422

Project title HUMAINE

Human-Machine Interaction Network on Emotions Contractual date of delivery November 30, 2005

Actual date of delivery November 30, 2005

Deliverable number D9d

Deliverable title Proposal for exemplars and work towards them:

Usability

Type Report

Number of pages 30

WP contributing to the

WP 9

deliverable

Task leader KTH

Author(s) See page 5

EC Project Officer Philippe Gelin

Address of lead author:

Kristina H??k

Department of Computer and Systems Sciences

Stockholm University/Royal Institute of Technology (KTH)

Forum 100

164 40 Kista

Sweden

Table of Contents

1THE PLACE OF THIS REPORT WITHIN HUMAINE (5)

2BRIEF OVERVIEW OF WORKPACKAGE 9 AND THE EXEMPLAR PROPOSAL (6)

2.1The field covered by Workpackage 9 (6)

2.2The research objectives (7)

2.2.1 Main elements of the exemplar (8)

2.2.2 How the elements link to each other (8)

2.2.3 How the elements link to other aspects of HUMAINE (9)

3THE PLANNED PROGRAM OF RESEARCH (10)

3.1Element 1: Criteria for usable affective interaction systems (10)

3.1.1 Main participants (10)

3.1.2 Main steps planned towards producing element 1 (10)

3.2Element 2: Evaluation metrics for criteria (11)

3.2.1 Main participants (11)

3.2.2 Main steps planned towards producing element 2 (11)

3.3Element 3: Existing user - centred methods for design and evaluation (12)

3.3.1 Main participants (12)

3.3.2 Main steps planned towards producing element 3 (12)

3.4Element 4: New methods for design and evaluation (14)

3.4.1 Main participants (14)

3.4.2 Main steps planned towards producing element 4 (14)

4RESEARCH PROGRESS TO DATE (17)

4.1Element 1: Criteria for usable affective interaction systems (17)

4.2Element 2: Evaluation metrics (17)

4.3Element 3: Existing user - centred methods for design and evaluation (19)

4.4Element 4: New methods for design and evaluation (22)

5CONCLUSION (27)

5.1Relation to the state of the art (27)

5.2Evidence of esteem (27)

6REFERENCES (29)

1The place of this report within HUMAINE

The HUMAINE Technical Annex identifies a common pattern that is followed by most of the project’s workpackages

The measure of success will be the ability to generate a piece of work in each of the areas which exemplifies how a key problem in the area can be solved in a principled way; and which also demonstrates how work focused on that area can integrate with work focused on the other areas. We call these pieces of work exemplars. The exact form of an exemplar is not prespecified: it may be a working system, but it might also be a well-developed design, or a representational system, or a method for user-centred design. (p 4)

To that end, each thematic group will work out a proposal for common action, embodied in one or more exemplars to be built during the second half of the funding period (p.16)

The process will begin with production by each thematic group of a review of key concepts achievements and problems in its thematic area; and drawn from the review, an assessment of the key development goals in the area. This review and assessment will be circulated to the whole network for discussion and comment, aimed both at building understanding of basic issues across areas, and at identifying the choices of goal that would be most likely let the different groups achieve complementary developments. That consultation phase will provide the basis for deliverables in month 11, which describe in some detail a few alternatives that might realistically be chosen as exemplars in each area, and their linkages to issues in other thematic areas. A decision and planning period will follow, involving consultation within and between thematic areas, leading to presentations at the second plenary conference, which will describe a single exemplar that has been chosen for development in each area, and the way work on the exemplar will be divided across institutions. The remainder of the project will be absorbed in developing the chosen exemplar. (p. 21)

The consultation phase has now ended. Near-final plans were presented to the whole network at the Plenary in May 2005, and adjustments have been made accordingly.

This deliverable reports the plans that have now been set out for the remaining 25 months of the project. They are necessarily provisional, because they will be subject to two reviews (in 2006 and 2007) before they are completed.

Work has begun on several aspects of the planned programme. It is also reported.

Ethical issues affect the whole of HUMAINE, but rather than repeating essentially similar points in multiple deliverables, they will be handled coherently in a single document, D0o (Science and Society).

The following persons have contributed to the work reported in the deliverable:

Kristina H??k, Jarmo Laaksolahti, Fiorella de Rosis, Brigitte Krenn, Katherine Isbister, Benoit Morel, Asimina Vasalou, Catherine Pelachaud, Markus Ballegooy, Stephen Westerman, Tanja B?nziger and WP9 members

The institutions that have contributed are:

KTH, OFAI, Imperial College of Science, Technology and Medicine, University of Paris 8, Università Degli Studi di Bari, T-Systems Nova GmbH, La Cantoche Production, LEEDS, GERG

2Brief overview of Workpackage 9 and the exemplar

proposal

While the rest of HUMAINE will produce knowledge and theories of many different aspects of emotional systems (ECA behaviours, emotion recognition, interaction principles, exemplar databases, and emotion theory) the overall goal of WP9 is to use that knowledge to produce design and evaluation methods able to deliver functioning affective end-user applications. Such applications allow users to become affectively involved in the course of interaction with the system.

Our strategy for working within an end-user application framework to evolve the usability of affective systems can be divided into two focus areas:

1.Ends: Determining qualities and criteria that demark emotional systems as usable and

evoking the desired experiences for users.

2.Means: Forging process—design methods, project goals and evaluation strategies —

that will steer a project towards producing a ‘successful’ affective application.

The WP9 exemplar is designed to develop, explore and determine key criteria and methods for designing and evaluating affective systems

2.1 The field covered by Workpackage 9

The area covered by this workpackage is described in the Technical Annex, particularly in Section 6.2, and in more depth in the review and assessment document for the workpackage. We summarise the area here so that the deliverable can be read as a stand-alone document. Regarding Ends, we do not believe there are simple, replicable correlations between system properties and all end users’ experiences. The experience of using an emotional system is not a property of the system itself, but rather is something that arises in the interaction between user and system. To quote Sengers and colleagues (2004):

“Rather than experience as something to be poured into passive users, we argue that users actively and individually construct meaningful human experiences around technology. They do so through a complex process of interpretation, in which users make sense of the system in the full context of their everyday experience.”

Along the same lines, Suchman has criticized the cognitivistic reductionist basis for the field of affective computing (Suchman, 2002), reducing human emotional responses to discrete, universal component parts, arising from some underlying state, and offers examples of alternative designs that put emotional interaction at the core but where: “we might differently conceptualise affective encounters at the interface: as irreducibly contingent meetings of particularly situated persons with equally particular, dynamic, and culturally inflected things.” As we discuss below, usability of emotional systems and the process of arriving at functioning emotional systems need to be sensitive to how end-users are part in co-constructing the experience and be part of this dynamic and cultural process.

Usability traditionally focuses on goals such as effectiveness, efficiency, safety, utility, learnability, and memorability. These objective usability goals contrast with user experience goals, which cover subjective qualities such as being fun, rewarding, motivating, satisfying, enjoyable, and helpful. Usability goals and user experience goals often stand in complex relationships, involving tradeoffs such as safety vs. fun or efficiency vs. enjoyability (Preece et al., 2002). Introducing emotion thus raises many new dimensions for research on usability to address. It becomes, for instance, a serious issue whether users feel a system is ‘sympathetic’ or morally acceptable, whether it engages them emotionally. Furthermore, emotions address directly inherently adaptive faculties of humans, posing challenges for methods of user studies and artefact design.

Work Package 9 will take a primarily qualitative, situated/contextual approach to measuring Ends. A sub-group of Work Package 9 may also focus on evaluating and developing measures aimed at isolating meaningful evaluative variables and testing various components of systems against these variables.

Regarding Means, we believe that evaluation of affective systems is vital not just at the end of the design process, but as an integral part of the design process from the beginning. Having the ability to bounce early intuitions and design sketches off of real users can make key contributions to the evolution of a truly engaging end application, and may even inform the affective theory that led to the application itself.

Work in the field of HCI has made great strides toward merging design and evaluation of productivity-oriented systems (see the goals section of this document for some relevant citations); we seek to extend and adapt participatory design methods and approaches to the needs and desired outcomes of affective system projects. For example, encouraging nonverbal participation and evaluation, and using physical prototypes and fake systems operated by humans. We believe this holistic and integrated approach will not only lead to better system designs, but may also provide important ‘in situ’ insights back to affective theory makers. Thus, it is not the primary intent of WP9 to test computational emotion theories or evaluate how particular parameters for e.g. raising an eyebrow in an ECA should work. Instead, the task of WP9 will be to produce design process and evaluation theory, methods, and measures needed to take all that knowledge into the design process and through it, to produce emotional systems that will engage with end-users to create compelling experiences.

2.2 The research objectives

The exemplar proposed for WP9 is a Framework for Design and Evaluation of Usable Affective Interaction Applications. Similar to how HCI has benefited from user-centred design processes, our aim is to test and further develop such methods for the area of affective interaction. This in turn will hopefully push the research frontier forwards and serve as an important exemplar to others in the field. The framework contains an assortment of techniques and methods for various aspects of the design process including generating initial ideas for affective applications, refining ideas into systems/products or validating the affective interaction loop in finished systems/products. As a whole the framework is intended to be a broad resource for designing and evaluating affective systems – representing first and foremost a user-centred perspective on affective interaction applications. The tools and methods of the framework will be applied and evaluated in a variety of situations/applications/domains. Records from the sessions will provide valuable guidance for future users of the tools and methods regarding their proper usage and expected results.

2.2.1Main elements of the exemplar

The exemplar consists of four main elements.

Criteria for usable affective interaction systems

Develop an understanding for what makes affective interaction systems successful and formulate some criteria. These criteria will not be objective, independently measurable entities, but will make sense relative to the specific application domain, aim to capture subjective experiences of the application, and foremost, be related to the designer’s intention for the application.

Evaluation metrics for criteria

Translate criteria for affective interaction systems into usable metrics and methods for eliciting them.

Existing user-centred methods for design and evaluation

Condense experiences from applying existing user-centred design methods to the design and evaluation of affective interaction systems – which methods work and which do not? Examples of such methods include body-storming (Oulasvirta et al, 2003), interaction re-labelling (Djajadiningrat, et al., 2000) and personas (Cooper, 1999).

New methods for design and evaluation

Develop new methods for capturing unique aspects of affective interaction that can be used during an iterative design-evaluate-redesign process. Methods that will be investigated include: a sensual method for non-verbal mediation of affective state, a Wizard-of-Oz environment for multimodal emotional interaction, and an extended think-aloud protocol designed to capture emotional interactions.

2.2.2How the elements link to each other

Design and evaluation is a broad area, participants have differing interests and backgrounds, different use-cases require different methods, resources, and study angles. We are interested in the whole design-evaluation process (see Figure 1) and the subtasks of WP9 address different aspects of the process. The exemplar can still be viewed as a single piece though, since it explores one and the same idea: that a user-centred perspective will help focus the design process and create final application systems that involve users affectively. This exemplar will be a touchstone for integrating this perspective throughout the whole life-cycle of a system development process. While we strongly believe that this is the best kind of exemplar for WP9, there are still many white spots on the map. We do not know how to capture subjective experiences and affective involvement in such a way that it can provide feedback into the design process. A user-centred design perspective and development philosophy is most probably a good path to explore, but little is, as of yet, known in what to design for (flow, pleasure, fun, experiences, excitement, fear), and how to best make users involved.

Figure 1User-centred design framework

Positioning the subtasks within the user-centred design framework above ensures that subtasks are working towards a common goal. The nature of the framework with its different methods and tools all sharing the same conceptual framework also helps in this respect. The four subtasks follow on one-another. Once criteria for the success of a specific application project are established, there has to be corresponding measurements to know whether the design goals are met according to the same criteria. The process to ensure such a design and evaluation cycle in turn needs methods. All these tasks together will help shape the framework produced by this WP.

2.2.3How the elements link to other aspects of HUMAINE

WP9 cannot be separated from the rest of HUMAINE because evaluation depends on systems (or concepts) to evaluate. It can be seen in section 4 that there has been sustained interaction with teams in most areas of the network. The WP9 workshop, which will take place in January 2006, is designed to strengthen these collaborations. The main kinds of interaction can be summarised briefly as follows.

WP3: Methods need to be grounded in theory. WP9 absorbs and builds upon current affective theory to create appropriate evaluation instruments.

WP5: The annotated databases are expected to provide input to design processes in WP9, and will also serve as material in evaluation processes.

WP6: By working with Work Package 6 in the design of an Affective Interactive Embodied Conversational Agent system, we hope to test and refine our evaluation techniques and instruments, which will serve as our final exemplars. Moreover we plan to apply the representation language defined in WP6 for the WoZ-environment and also adopt it to the “sensual” method.

WP8: We may start a similar collaboration as with WP6 on specific application examples where we can apply our framework for design and evaluation.

3The planned program of research

3.1 Element 1: Criteria for usable affective interaction systems

3.1.1Main participants

Institution Person(s)

OFAI Brigitte Krenn

Università Degli Studi di Bari Fiorella de Rosis

KTH Kristina H??k, Jarmo Laaksolahti

3.1.2Main steps planned towards producing element 1

Subtask Carried out by Start / end

dates

OFAI, BARI M0/M48 Compilation of a commented literature list on ECA

evaluation studies and of initial definition criteria for

empathic ECAs.

KTH M0/M48 Explore success criteria for affective interaction wrt the

specific design aims.

The first task aims at a commented compilation of literature on ECA evaluation. As a starting point, the issue is addressed from three different perspectives: 1) Dimensions of evaluation, i.e. what aspects or conceptual dimensions of ECAs are evaluated, for instance: ECAs as social actors, ECAs as emotion expressing entities, ECA applications and their emotional effect on the user, etc. 2) Evaluation paradigms addressing the type of study, for instance: 3rd party observation, i.e. the user/subject is only observer of pre-fabricated scenes played by one or more ECAs; Wizard-of-Oz, i.e. the system is merely mock-up and the system behaviour is controlled by a human (the wizard); full user involvement with a fully automatic system, i.e. the user interacts with a fully computerized system. 3) Evaluation methods: While 1) and 2) address macro-levels of evaluation, 3) is concerned with micro-level aspects of evaluation, i.e. the individual strategies and methods employed in different evaluation experiments, such as quantitative and qualitative methods, questionnaires, semantic differentials, the specific set up of study plans and the such.

This contribution initially has been conceived as a literature survey terminating at month 19. In the course of the work in HUMAINE it has turned out that the task is better conceived as work in progress which is open during the runtime of the network and ideally stays active beyond its runtime, reflecting, complementing and continuing the work in HUMAINE.

In the second task we aim to explore success criteria for affective interaction application wrt the specific design aims of an application and what the designer intended to achieve. Only if the designer’s intentions are experienced by the end-users as intended is it possible to talk about a successful system. In addition more general concepts like flow, ambiguity, pleasure and expressivity will be explored as potential criteria for success. This will mainly by accomplished through a series of workshops, brainstorming sessions and practical application to use cases (se below). Our aim is that this will lead to an iterative refinement of useful criteria for developing affective interaction systems. A first workshop was held at the CHI conference in April 2005 and will be followed by a brainstorming session and the WP9 workshop in January 2006.

3.2 Element 2: Evaluation metrics for criteria

3.2.1Main participants

Institution Person(s)

KTH Kristina H??k, Jarmo Laaksolahti

OFAI Brigitte Krenn

Università Degli Studi di Bari Fiorella de Rosis

3.2.2Main steps planned towards producing element 2

Subtask Carried out by Start / end

dates

OFAI/BARI M13/M48 Construction of a collection of design, test and evaluation

strategies and methods for empathic ECAs.

Assessment of subjective evaluation metrics KTH M0/M30 The first task aims to give a review of the state-of-the art in evaluation studies on ECAs. In particular, the review will start from a reflection on the concept of ‘empathy’, to critically reflect on whether and how ECAs have proven, so far, to be able to establish this kind of relationship with their human users. The study will consider the evaluation criteria and methods applied, and will critically present in a flexible and dynamic structure the results produced. Results from this task feeds into “Criteria for affective interaction systems” and “Existing user-centred methods for design and evaluation”. The studied literature will also appear in the commented literature list described in task 1 under 3.1.2.

The second task deals with subjective evaluation metrics. A subjective evaluation metric can be quite open-ended – any sources of information may be collected and interpreted vis-à-vis the goals of the system. Anecdotal evidence, informal chats between users and system-

builders, tiny study sizes, forms structured to influence user interpretation, no discussion or analysis of results: this may sound like a to-do list for bad evaluation. But these choices are deliberately and thoughtfully made, highlighting underlying conceptual problems in using standard so-called ‘scientific’ evaluation techniques to evaluate emotionally involving systems. The subjective approach means that we might not be building systems for ‘normal’ or ‘average’ users. Instead we are interested in the richness and complexity of unique, individual users, cultural contexts, and resulting variety of interpretations and experiences of our systems. It is not appropriate to summarize the results of a study into a few statements that are said to hold for everyone. Also, the statistical averaging and laboratory simplifications necessary for reliable scientific statements may wash out all the details that interest us in a user-centred design process where we want input for the next cycle of design. We may prefer a rich, narrative, and singular understanding to a simpler but rigorous and generalisable understanding in the early stages of design. The eMoto study and the Sensual Evaluation Instrument (see below) are examples of an evaluation study and - method respectively where subjective metrics are essential.

Subjective measurements will be complemented by more generalisable and objective measures where appropriate and needed in order to further our knowledge of what constitutes a successful design. This will mainly by accomplished through a series of workshops, brainstorming sessions and practical application to use cases (se below). Our aim is that this will lead to an iterative refinement of useful metrics for evaluating affective interaction systems. A first workshop was held at the CHI conference in April 2005 and will be followed by a brainstorming session and the WP9 workshop in January 2006.

3.3 Element 3: Existing user - centred methods for design and

evaluation

3.3.1Main participants

Institution Person(s)

La Cantoche Production Benoit Morel, Justine Plessier

T-Systems Nova GmbH Markus Ballegooy

University of Paris 8 Catherine Pelachaud

KTH Kristina H??k, Jarmo Laaksolahti

LEEDS Stephen Westerman

3.3.2Main steps planned towards producing element 3

Subtask Carried out by Start / end

dates

WP9 Brainstorming session on generating application ideas using existing design and evaluation methods KTH, Paris 8, Cantoche, OFAI

M25/M25

Application of existing user-centred methods to use-cases KTH, Paris 8,

Cantoche, T-

Systems

M0/M48

Develop a taxonomy of affective systems usability testing LEEDS M24/M33 In conjunction to the WP9 workshop to be held in Stockholm, a brainstorming session will be organised. The session will be devoted to exploring how to use existing methods for design and evaluation to generate usable affective application ideas. Participants will include industrial as well as academic partners from HUMAINE.

Throughout the project methods for user-centred design and evaluation will be applied to the use-cases proposed in D9c (e.g. the eMoto application described below), as well as other applications, and experiences will be gained regarding their suitability for affective interaction design. For instance, until now, for every application to be developed, Cantoche meets their clients in order to understand why an ECA should be used and what is the most appropriate one. Together, they define:

- The mission for the ECA ( e.g. Expert, tutor, assistant).

- The target: what kind of users will interact with the ECA (culture, gender, age)

- The scene: what is the support where the ECA will act (mail, web site, presentation) All along the development of the application, the client tests the system and provides feedback that helps to refine it before the application is launched to real users. So, our first users are the clients. Within WP9, we aim to bring na?ve users in the development process and evaluate what users centred methods can bring us.

Within WP6, Paris 8 works on capabilities required to build an affective interactive ECA. Some cover low level aspects of ECA specification (behaviour expressivity factor, multimodalities coordination) other deal with the definition of means to show level of engagement through displaying backchannels, or to endow agent with visual attention and perception. While in WP6 we are concentrating in defining the concepts and providing some forms of implementation of these capabilities, no work is undertaken to understand if and when these capabilities are necessary and important for a given application. For example, we would like to know to which degree of subtlety ECA behaviours should be specified: e.g., to which degree it is important for the application to have an ECA able to display facial expressions of blend of emotions. We aim to use the outcome of user centred studies to understanding these issues and be able to refine which capabilities are required to build an affective interactive ECA in the context of an application.

LEEDS will develop a simple taxonomy of affective systems that can be used as a basis for the examination of usability issues. This taxonomy classifies on the basis of direction of emotional communication and requirements for elicitation and assessment of user affect. Existing systems and empirical evidence will be reviewed in this context, with specific reference to usability-related outcomes, and implications for ‘traditional’ usability testing methods.

3.4 Element 4: New methods for design and evaluation

3.4.1Main participants

Institution Person(s)

KTH Kristina H??k, Jarmo Laaksolahti, Katherine

Isbister

OFAI Brigitte Krenn

Università Degli Studi di Bari Fiorella de Rosis

Imperial College Asimina Vasalou

LEEDS Stephen Westerman

3.4.2Main steps planned towards producing element 4

Subtask Carried out by Start / end

dates Sensual evaluation instrument KTH M0/M48 - Design of the sensual evaluation instrument KTH M0/M36 - Application/evaluation of the instrument KTH M18/M48 The first task will investigate the possibilities for subjective self-report of affective state in nonverbal ways. As described above this work has already started and resulted in one possible toolset. This toolset will iteratively be refined and expanded during the project.

The second task deals with applying the toolset in “real-world” settings to verify its functionality and to gain a better understanding of possibilities and limitations of the proposed form of evaluation studies. Currently a version of the toolset has been constructed. In the next phase of development, we plan to use the Sensual Evaluation Instrument with designers who are seeking affective feedback from users about their work, with preference given to HUMAINE members' projects. We will include the SEI in the WP9 Workshop in January in Stockholm as a tool for consideration and use. We hope that testing the tool as a design aid with evolving systems will validate the SEI's helpfulness, as well as lead to evolution of the tool itself.

Subtask Carried out by Start / end

dates

WoZ test bed OFAI/BARI M0/M48

- Implementation of the WoZ tool BARI M0/M19

- Realizations of the design-evaluation loop OFAI/BARI M13/M48

The first task aims to produce a tool which enables performing evaluation studies on ECAs in different conditions: different ECAs, different domains, and different evaluation criteria. This tool will enable performing a comparison of results that may be obtained by means of ‘subjective’ (or cognitively-based) methods and by means of ‘behavior observation’ methods: therefore, a reflection on ‘methods’ and evaluation criteria’.

The second task concerns design and conduction of WoZ experiments, in order to hypothesize and test aspects of empathic interactions of users with ECAs. These experiments will consist in sets of small-scale iterative studies, which will be designed so as to provide to users a framework which will enable them to express their requirements with a relatively high degree of freedom. Results expected from these experiments will not only consist in statistics about subjective and objective evaluation parameters, but will provide, as well, a corpus of data upon which to ground the design of artificial dialogs. The insights gained from every WoZ experiment will therefore feed into the redesign of the WoZ experiments, as well as into modelling of fully computer generated empathic ECAs, the effect of which will then be tested in user experiments. Results from this task feeds into “Evaluation metrics for criteria” and “Exploring new methods for designs and evaluation”. OFAI, in particular, will make use of the WOZ setting in order to explore and develop methods and strategies suitable for rapid prototyping of an application where a human user and an ECA collaborate in a quiz game. This work will be done in the context of an FP6 Cognitive Systems project that is about to start beginning of 2006.

Subtask Carried out by Start / end

dates

Extended think aloud protocol Imperial, GERG M27/M31

- Development of affective applications Imperial, GERG M27/M28

Imperial, GERG M28/M29 - Development of the extended emotional-think aloud

protocol and pilot study

- Cross Cultural think aloud study Imperial, GERG M29/M31

We are currently developing an affective application that will serve as a platform for our experiment. Our objective is to determine whether the think aloud protocol can reveal emotional data. For the purposes of our study, we are developing an application that will support a short set of tasks that include three emotion inducing events, one for joy, one for anger and one for embarrassment.

We will pilot a few think aloud prompts. In one concept we will prompt users with a flickering light which will light up their screen when they are silent for too long. In another concept we will inform users that the facilitator is prompted by the system on silent moments and speaks out during those moments. What we hope to gain from this pilot is an

understanding on which concept is more personable and less alienating to the user. We will also use questionnaires to access to which extent the 2 criteria of interference and suppression are being affected.

Once we choose a prompting system we will conduct the experiment. In this experiment, each user will be trained on the think aloud protocol. Following training, he/she will interact with each of the three emotion-inducing applications. The verbalisations will be recorded and then analysed and rated by na?ve, independent experts. The study will be conducted in the UK and in Switzerland in anticipation and detection of cultural differences in vocal expression. We do not expect our sample to exceed 30 users (15 for each country).

Subtask Carried out by Start / end

dates Psychometric assessment model/approach LEEDS M24/M30 - Further testing of the model LEEDS M24/M28 - Revision of the model LEEDS M28/M30 LEEDS will be beginning further tests of their model/assessment approach in month 24. Results relevant to HUMAINE are expected to be available in month 28. Further revisions to the model/assessment method are expected to be available in month 30.

4Research progress to date

4.1 Element 1: Criteria for usable affective interaction systems Progress on the “webpage consisting of a commented literature list on ECA evaluation studies”: The literature that has been collected so far is considered as a seed for a “literature collection in progress”. A first version of the commented and structured list will be made available on the HUMAINE portal by the end of 200

5. For a brief description of the initial structure of the collection see section 3.1.2. This list will be open for additions of literature as well as supplementary comments on existing entries and alternative structuring of the entries constituting the seed. Ideally a single literature entry will be embedded in a web of links and be annotated with a number of comments referring to different aspects of ECA evaluation. Exploring criteria for affective interaction systems is mainly accomplished through a series of workshops, brainstorming sessions and practical application to use cases. Our aim is that this will lead to an iterative refinement of useful criteria for developing affective interaction systems. A first step was the workshop on evaluation of affective interfaces was organised at the CHI (Computer-Human Interaction) conference in Portland, Oregon. The workshop was named “Evaluating Affective Interfaces: Innovative Approaches” and 12 papers were selected for presentation at the workshop. The researchers at the workshop presented work that spanned a large range of theoretical perspectives and thereby quite a large span of possible methods for evaluation. In our perspective, this shows the diversity of what is meant by affective interactive systems and also how different researchers view what can be seen as successful design/criteria. The CHI workshop will be followed up by the WP9 workshop in January 200

6. Prior to the workshop a brainstorming session will be held that explores how to generate application ideas using existing design and evaluation methods. The session and the following workshop will include researchers, designers and practitioners. These groups all have their own views on what is important in a design process. For researchers the most important thing is to prove the success of their systems using some suitable (usability)criteria. Designers on the other hand search for methods that support and provide feedback to the design process. Practitioners search for good examples that illustrate ideas in action and that can provide practical advice on how to build applications. The list of problems collected for deliverable D9a also shows the diversity of possible criteria. In order to gain a complete picture of affective interaction design and evaluation, all of the above viewpoints need to be taken into account and their relationships explored.

4.2 Element 2: Evaluation metrics

One of the main goals of HUMAINE is to make different theoretical strands within the large field of affective computing to get together and debate their different viewpoints. The aim is to enable fruitful new collaborations between fields that may previously have been more or less unaware of each other. In WP9 one such theoretical stance debate has been brought up several times. In short summary it is a meeting between Human-Computer Interaction goals and methods and more psychology-inspired goals and methods. For example, we debate the purpose behind why we do user studies, how to do them, and which methods are "scientific" and which are not.

In HCI, researchers are turning more and more towards what we could name "ecological validity". That is, systems needs to be brainstormed, developed and tested in settings that are

very close to the "real" settings they will be used in. We talk about studies "in the wild" and issues like "staged lived experiences", that is, how we can set up situations with half-working systems that resemble the real situations in which they will be used.

Theoretical foundations from this movement is picked from phenomenology (see e g Paul Dourish' book "Where the action is"), from ethnographic work, from grounded theory, from situated cognition (see Suchman "Plans and situated action"), from activity theory (see Nardi "Context and Consciousness: Activity Theory and Human-Computer Interaction”), participatory design (or the so-called Scandinavian school of design), and so on. In common for all these is an emphasis on moving beyond cognitivism and instead try to see the whole context in which a system is used, the other tools people make use of, their daily practices, their culture, etc. The belief is by understanding more of the whole complexity and being more “emphatic” with the end users, actually placing ourselves as designers in their shoes so to speak, we will be able to produce better systems.

In our field, affective interactive systems, we have additional problems compared to more traditional HCI-problems with the development with tools. We are trying to design for some kind of user experience. We cannot guarantee that the users will experience something because it is obviously constructed by themselves, in their minds, but we can design more or less good interaction that at least form the basis for certain kinds of experiences. All we can do is a more or less good job in the design process. The problem lies in how we evaluate our systems so that we can provide the designer of the system with the right level of feedback? What will help the designer to figure out where the weaknesses in the system are and what the best way of altering it is to actually reach the design goals? As the experience has elements of unconscious processes (both cognitive and bodily) we need to find some way of tapping into those. In addition, we also need to compare those unconscious emotional reactions to the more conscious reflections that users make. Both are relevant in the design process. I will not buy a system that does not fit with my attitudes and beliefs about the system.

We therefore need two kinds of criteria: what are the reasons that customers will pick an application off the shelf in the store or on the web and decide to use it? And second, once they do indeed use it, what are the criteria for measuring that their reactions to it are indeed the ones that the designer aimed for?

In the more psychology-oriented methods, it becomes vital to do more controlled lab studies in order to single out which variables do indeed contribute to the success of an interaction with an affective interactive system. They typically require statistically significant results and ways of isolating what gives rise to what.

Most cognitivists, psychologists, lab-researchers are well aware of validity issues in general, including ecological validity. But they are facing a trade-off between preserving the complexity of "ecological" situations, on one hand, and experimental control, on the other hand. It is not infrequent that "ecological validity" (or rather complexity) is sacrificed for the benefit of studying specific behaviours in isolation. The inconveniences of isolating any psychological process from its context are well known, nevertheless it is often impossible to test hypotheses or make systematic observations in fluctuating contexts that might differentially affect the processes under investigation. Aspects of the context that are likely to influence the investigated processes must be controlled and/or systematically manipulated to allow researchers to understand their effects.

The problem for the emotion researcher lies in the following: Emotions are reactions that include several components (facial/vocal/gestural/postural expressions, physiological

reactions, feelings, they are tightly linked to attentional and motivational processes, memory, decision making, etc…). One important component is the subjective feeling associated to the emotional reaction. It belongs to the domain of the subjective experience of the person and can only be accessed through subjective reports. We have ways to measure the other components of the emotional reactions without asking the person to give a report, but if we want to know what they feel we can relay only on subjective reports (there is no alternative method).

Both these viewpoints and methodological perspectives need to co-exist side by side in the work in WP9, and they can indeed inform each other, but we also need to recognise that they come from quite different theoretical foundations and will therefore contribute to our knowledge of what designs work in quite different ways.

A major problem in HUMAINE is that the different WPs have quite different goals. The goal of, for example, WP3 is to generate emotion theory in general. The goal of WP9 is to find the methods that will generate good applications. These are quite different goals. And they require different methods. An evaluation method in WP9 by necessity has a range of extra demands that it needs to fulfil, such as being cost-efficient, not avoiding the complexity of the future use situation, etc. Otherwise it will never be used by the people who are supposed to develop affective interactive applications in the future.

Thus, the discussion on evaluation metrics is in a very lively state in WP9 where these two theoretical strands meet. One strand argues that isolating variables in the lab is crucial in order to know what we are measuring and whether indeed our affective interactive systems form the basis of the kinds of experience we want users to experience. Another strand is more interested in a phenomenological strand where evaluation is seen as an on-going process between users and designers where the whole story needs to be taken into account. We are certain that there will be a lively debate at the upcoming WP9 workshop on these issues.

4.3 Element 3: Existing user - centred methods for design and

evaluation

Several evaluation studies have been performed, or are planned, that explore the use of existing methods for user-centred design albeit sometimes in novel ways. Work has also been started by LEEDS on a taxonomy of affective systems that can be used as a basis for the examination of usability issues.

A wild evaluation of users’ emotional engagement

eMoto is a mobile service for sending and receiving affective messages, with the explicit aim of addressing the inner experience of emotions. In eMoto a user first writes her text message and then finds a suitable affective graphical expression to add to the background of her text. To find these expressions, she navigates in a circular background of colors, shapes and animations through using a set of affective gestures. Following the two-tiered evaluation model (H??k, 2004) we first verified that users understood and could interpret each of these parts of eMoto before we combined them and set up a user study where we looked for results on the overall goal, which was having users emotionally engaged in an affective loop.

Figure 2 Users' probes

Figure 3 Friends' probes To determine whether someone is emotionally engaged and moreover emotionally involved in an affective loop, is however a very difficult task. In our final evaluation of eMoto we went for a combination of Isomurso and colleagues’ following a friend method (Isomurso et al. 2004) and the cultural probe method first described by Gaver and colleagues (Gaver et al. 1999). Both these methods contain aspects of letting users themselves become the researchers. In this final evaluation we wanted users to experience eMoto in the wild and not in the more staged lab environment where we believe it is hard for us to create a setting where users can become truly emotionally engaged, but also the wild, would not be that much wild, if we researchers came along. Instead we gave users material for them themselves to document their experiences (see Figure 2), but also we prepared a bag of material for them to give to a close friend of theirs (see Figure 3). eMoto is very much inspired by emotional body language, a language in which friends understand each other far better than we as outsiders understand them. Therefore we believe friends can be better, than at least computer scientists, performing the task of analyzing users’ emotional engagement.

After having received all the material from the users we quickly looked through all the films, diaries and postcards from both users and their friends and together with our log data formed a quick and dirty analysis of each user’s experiences. We then discussed this analysis with our users during individual interviews. The intention was to get a starting point for the discussion but still stay to the notion of letting the users become the researchers.

We have just started to go through these interviews but preliminary results indicate that we in our design of eMoto have succeeded in presenting users with a foundation of expressiveness that carries shared meaning but opens up for individual expressiveness and emotional engagement.

Evaluation of an emotion aware voice portal

One of the major problems in evaluating affective systems is to produce or find usage situations where real user emotions emerge spontaneously without an artificial add on (artificial laboratory situations). In T-System’s evaluation study we try to obtain “quasi – natural” usage scenarios by deceiving the subjects about the tests purpose.

SUSY is an emotion aware voice portal where customers can extend their contract with the operator and select a new mobile phone from a list. The usability of the complex selection process is the main subject of the study, also the general acceptability of “speech directed upgrading” is a question we want to clarify. Since there are many ways to get into problems during the selection process, we added an emotional component to the system. Within dialog situations where the user gets angry and anger is detected by the acoustic classifier the system

中考几何最值问题(含答案)

几何最值问题 一.选择题(共6小题) 1.(2015?孝感一模)如图,已知等边△ABC的边长为6,点D为AC的中点,点E为BC的中点,点P为BD上一点,则PE+PC的最小值为() 3 AE==3, . 2.(2014?鄂城区校级模拟)如图,在直角坐标系中有线段AB,AB=50cm,A、B到x轴的距离分别为10cm和40cm,B点到y轴的距离为30cm,现在在x轴、y轴上分别有动点P、Q,当四边形PABQ的周长最短时,则这个值为() 5050+50

LN=AS==40 MN==50 MN=MQ+QP+PN=BQ+QP+AP=50 =50 3.(2014秋?贵港期末)如图,AB⊥BC,AD⊥DC,∠BAD=110°,在BC、CD上分别找一点M、N,当△AMN周长最小时,∠MAN的度数为()

4.(2014?无锡模拟)如图,∠MON=90°,矩形ABCD的顶点A,B分别在OM、ON上,当B 在边ON上运动时,A随之在边OM上运动,矩形ABCD的形状保持不变,其中AB=2,BC=.运动过程中,当点D到点O的距离最大时,OA长度为() C OE=AE=AB=× AD=BC= DE= ADE==, =

DF=, OA=AD= 5.(2015?鞍山一模)如图,正方形ABCD的边长为4,点E在边BC上且CE=1,长为的线段MN在AC上运动,当四边形BMNE的周长最小时,则tan∠MBC的值是() C D ,连结,此时四 ,连结MN= =, =, ,

PC= PDC==. 6.(2015?江干区一模)如图,△ABC中,CA=CB,AB=6,CD=4,E是高线CD的中点,以CE 为半径⊙C.G是⊙C上一动点,P是AG中点,则DP的最大值为() C BG AD=BD=AB=3 CE=

幅度调制与相位调制

幅度/相位调制 过去几十年随着数字信号处理技术与硬件水平的发展,数字收发器性价比已远远高于模拟收发器,如成本更低,速度更快,效率更高。更重要的是数字调制比模拟调制有更多优点,如高频谱效率,强纠错能力,抗信道失真以及更好的保密性。正是因为这些原因,目前使用的无线通信系统都是数字系统。 数字调制和解调的目的就是将信息以比特形式(0/1)通过信道从发送机传输到接收机。数字调制方式主要分为两类:1)幅度/相位调制和2)频率调制。两类调制方式分别又成为线性调制和非线性调制,在优劣势上也各有不同,因此,调制方式的选择最终还需要取决于多方面的最佳权衡。 本文就对幅度/相位调制加以讨论,全文整体思路如下: 1 信号空间分析 在路径损耗与阴影衰落中已提出发送信号与接收信号的模型以复信号的实部来表示,而在本文中为了便于分析各调制解调技术,我们必须引入信号的几何表示。 数字调制将信号比特映射为几种可能的发送信号之一,因此,接收机需要对各个可能的发送信号做比较,从而找出最接近的作为检测结果。为此我们需要一个度量来反映信号间的距离,即将信号投影到一组基函数上,将信号波形与向量一一对应,这样就可以利用向量空间中的距离概念来比较信号间的距离。 1.1 信号的几何表示 向量空间中各向量可由其基向量表示,而在无线通信中,我们也可把信号用其相应的基函数来表示。本文我们讨论的幅度/相位调制的基函数就是由正弦和余弦函数组成的: 21()()cos (2)c t g t f t φπ=(1) 22()()sin (2)c t g t f t φπ=(2) 其中g (t )是为了保证正交性,即保证 220()cos (2)1T c g t f t dt π=? (3) 20()cos(2)sin(2)0T c c g t f t f t dt ππ=? (4) 则信号可表示为 12()()cos(2)()sin(2)i i c i c s t s g t f t s g t f t ππ=+ (5) 则向量s i =[s i1,s i2]T 便构成了信号s i (t )的信号星座点,所有的星座点构成信号星座图,我们把信号s i (t )用其星座点s i 表示的方法就叫做信号的几何表示。而两个星座点s i 和s k 之间的距离就是采用向量中长度的定义,这里不再赘述。 2 幅度/相位调制 相位/幅度调制主要分为3种: 1)脉冲幅度调制(MPAM):只有幅度携带信息;

7-7-5 容斥原理之最值问题.教师版

1. 了解容斥原理二量重叠和三量重叠的内容; 2. 掌握容斥原理的在组合计数等各个方面的应用. 一、两量重叠问题 在一些计数问题中,经常遇到有关集合元素个数的计算.求两个集合并集的元素的个数,不能简单地把两个集合的元素个数相加,而要从两个集合个数之和中减去重复计算的元素个数,即减去交集的元素个数,用式子可表示成:A B A B A B =+-(其中符号“”读作“并”,相当于中文“和”或者“或”的意思;符号“”读作“交”,相当于中文“且”的意思.)则称这一公式为包含与排除原理,简称容斥原理.图示如下:A 表示小圆部分,B 表示大圆部分,C 表示大圆与小圆的公共部分,记为:A B ,即阴影面积.图示如下:A 表示小圆部分,B 表示大圆部分, C 表示大圆与小圆的公共部分,记为:A B ,即阴影面积. 包含与排除原理告诉我们,要计算两个集合A B 、的并集A B 的元素的个数,可分以下两步进行: 第一步:分别计算集合A B 、的元素个数,然后加起来,即先求A B +(意思是把A B 、的一切元素都“包含”进 来,加在一起); 第二步:从上面的和中减去交集的元素个数,即减去C A B =(意思是“排除”了重复计算的元素个数). 二、三量重叠问题 A 类、 B 类与 C 类元素个数的总和A =类元素的个数B +类元素个数C +类元素个数-既是A 类又是B 类的元素个数-既是B 类又是C 类的元素个数-既是A 类又是C 类的元素个数+同时是A 类、B 类、C 类的元素个数.用符号表示为:A B C A B C A B B C A C A B C =++---+.图示如下: 教学目标 知识要点 7-7-5.容斥原理之最值问题 1.先包含——A B + 重叠部分A B 计算了2次,多加了1次; 图中小圆表示A 的元素的个数,中圆表示B 的元素的个数, 1.先包含:A B C ++ 重叠部分A B 、B C 、C A 重叠了2次, 多加了1次. 2.再排除:A B C A B B C A C ++---

容斥原理之最值问题

7-7-5.容斥原理之最值问题 教学目标 1.了解容斥原理二量重叠和三量重叠的内容; 2.掌握容斥原理的在组合计数等各个方面的应用. 知识要点 一、两量重叠问题 在一些计数问题中,经常遇到有关集合元素个数的计算.求两个集合并集的元素的个数,不能简单地把两个集合的元素个数相加,而要从两个集合个数之和中减去重复计算的元素个数,即减去交集的元素个数,用式子可表示成:A U B=A+B-A I B(其中符号“U”读作“并”,相当于中文“和”或者“或”的意思;符号“I”读作“交”,相当于中文“且”的意思.)则称这一公式为包含与排除原理,简称容斥原理.图示如下:A表示小圆部分,B表示大圆部分,C表示大圆与小圆的公共部分,记为:A I B,即阴影面积.图示如下:A表示小圆部分,B表示大圆部分,C表示大圆与小圆的公共部分,记为:A I B,即阴影面积. 1.先包含——A+B 重叠部分A I B计算了2次,多加了1次; 包含与排除原理告诉我们,要计算两个集合A、B的并集A U B的元素的个数,可分以下两步进行: 第一步:分别计算集合A、B的元素个数,然后加起来,即先求A+B(意思是把A、B的一切元素都“包含” 进来,加在一起); 第二步:从上面的和中减去交集的元素个数,即减去C=A I B(意思是“排除”了重复计算的元素个数).二、三量重叠问题 A类、B类与C类元素个数的总和=A类元素的个数+B类元素个数+C类元素个数-既是A类又是B类的元素个数-既是B类又是C类的元素个数-既是A类又是C类的元素个数+同时是A类、B类、C类的元素个数.用符号表示为:A U B U C=A+B+C-A I B-B I C-A I C+A I B I C.图示如下:

小学奥数——最值问题

最值问题 一、例题讲解 例题1.把1、2、3……16分别填进图中16个三角形里,使每边上的7个小三角形里的数之和相等,问:这个和的最大值是多少? 例题2.一把钥匙只能开一把锁,现在有9把钥匙和9把锁,但不知道哪把钥匙开哪把锁,那么最多要试开多少次才能配对好全部的钥匙和锁? 例题3.一种购物券的面值只有1元、3元、5元、7元和9元五种,为了直接付清1元、2元、3元……98元、99元、100元各种整数元的物品,至少要准备多少张什么样的购物券? 例题4.猴妈妈摘来一筐桃,将它们三等分之后还剩2个桃;取出2份,将它们三等分之后还剩2个桃;再取出2份,将它们三等分之后还剩2个桃 .猴妈妈至少摘了几个桃? 例题5.a 和b 是小于100的两个不同的自然数(非0),那么, 的最大值是多少? 例题6.把14拆成几个自然数相加的形式,再把拆成的这些自然数乘起来,如果想让所乘的积最大,应该怎么拆? 例题7.10、9、8、7、6、5、4、3、2、1这十个数,在每相邻的两个数之间添上一个加号或一个减号,组成一个算式,使它符合下面两个要求:① 算式的结果等于37,② 这个算式中所有的减数(前面添了减号的数)的乘积尽可能的大,那么这些减数的最大乘积是多少? 例题8.三个连续的自然数,后面两个数的积与前面两个数的积之差是114,这三个数中,最小的数是多少? 例题9.有三个数字,能组成6个不同的三位数,这6个数的和是2886,那么所组成的6个三位数中,最小的三位数是几? a - b a + b

二、举一反三 ① 如果四个人的平均年龄是30岁,并且在这四个人中没有小于21岁的,那么年龄最大的人是几岁? ② 将5、6、7、8、9、10六个数分别填入下面的圆圈内,使三角形每边上三个数的和相等,这个和最大是多少? ③ 现在有1元、2元、5元和10元的人民币若干张,如果要付清1元、2元、3元……98元、99元、100元各种整数元,至少需要准备多少张什么样的人民币? ④ x 和y 是小于50的两个不同的自然数(非0),并且x >y ,那么, 的最小值是多少? ⑤ 把50拆成几个自然数的和,要使这些自然数的乘积最大,应该怎么拆? ⑥ a 、b 、c 是从大到小排列的三个数,且a – b = b – c ,前两个数的积与后两个数的积之差是280,如果b = 35,那么c 是多少? ⑦ 把135个苹果分成若干份,使任意两份的苹果数都不相同,最多可以分成多少份? ⑧ 育才小学六(1)班51名学生一共植树251棵,已知植树最少的同学植树3棵,又知最多有11名同学植树的棵树相同,植树最多的同学可能植树多少棵? ⑨ 有一架天平,只有5克和30克的砝码各一个,现在要把300克的盐分成3等份,那么最少需要称几次? ⑩ 一个布袋里有大小相同、颜色不同的一些小球,其中红的10个、白的9个、黄的8个、蓝的2个,一次最少取多少个球,才能保证有4个球颜色相同? x - y x + y

幅值裕量和相位裕量

一般来说,)(ωj G 的轨迹越接近与包围-1+j001j +-点,系统响应的震荡性越大。因此,)(ωj G 的轨迹对01j +-点的靠近程度,可以用来度量稳定裕量(对条件稳定系统不适用)。在实际系统中常用相位裕量和增益裕量表示。 Re Positive Phase Margin Negative Gain Margin Negative Stable System Unstable System (ωj G

64 ω Log ω Log ω Log ω Log ?-90? -270?-180Positive Gain Margin Positive Phase Margin Negative Gain Margin Negative Phase Margin Stable System Unstable System dB ? -90? -270?-1800 dB 图1 稳定系统和不稳定系统的相位裕度和幅值裕度 相位裕度、相角裕度(Phase Margin)γ 设系统的截止频率(Gain cross-over frequency)为c ω 1)()()(==c c c j H j G j A ωωω 定义相角裕度为 )()(180c c j H j G ωωγ+?= 相角裕度的含义是,对于闭环稳定系统,如果开环相频特性再滞后γ度,则系统将变为临界稳定。 当0>γ 时,相位裕量相位裕度为正值;当0<γ时,相位裕度为负值。为了使最小相位系统稳定,相位裕度必须为正。在极坐标图上的临界点为0分

贝和-180度。?-180 增益裕度、幅值裕度(Gain Margin)h 设系统的穿越频率(Phase cross-over frequency) πωωω?)12()()()(+== k j H j G x x x ,Λ,1,0±=k 定义幅值裕度为 ) ()(1 x x j H j G h ωω= 幅值裕度h 的含义是,对于闭环稳定系统,如果系统开环幅频特性再增大h 倍,则系统将变为临界稳定状态。 若以分贝表示,则有 )()(log 20)(x x j H j G dB h ωω-= 当增益裕度以分贝表示时,如果1>h ,则0)(>dB h 增益裕度为正值;如果1

容斥原理之最值问题

教学目标 1. 了解容斥原理二量重叠和三量重叠的内容; 2. 掌握容斥原理的在组合计数等各个方面的应用. 知识要点 一、两量重叠问题 在一些计数问题中,经常遇到有关集合元素个数的计算?求两个集合并集的元素的个数,不能简单地把 两个集合的元素个数相加,而要从两个集合个数之和中减去重复计算的元素个数,即减去交集的元素个数, 用式子可表示成: AUB A B AI B (其中符号“ U ”读作“并”,相当于中文“和”或者“或”的意思; 符号“ I ”读作“交”,相当于中文“且”的意思. )则称这一公式为包含与排除原理,简称容斥原理?图示 AI B ,即阴影面积?图示 第一步:分别计算集合 A 、B 的元素个数,然后加起来,即先求 A B (意思是把A B 的一切元素都“包含” 进来,加在一起); 第二步:从上面的和中减去交集的元素个数,即减去 C AI B (意思是“排除”了重复计算的元素个数 )? 、三量重叠问题 A 类、 B 类与 C 类元素个数的总和 A 类元素的个数 B 类元素个数 C 类元素个数 既是A 类又是B 类 的元素个数 既是B 类又是C 类的元素个数 既是A 类又是C 类的元素个数 同时是A 类、B 类、C 类的元 素个数.用符号表示为: AUBUC A B C AI B BI C AI C AI BI C .图示如下: 如下:A 表示小圆部分, B 表示大圆部分, C 表示大圆与小圆的公共部分,记为: 包含与排除原理告诉我们,要计算两个集合 A B 的并集AU B 的元素的个数,可分以下两步进行:

例题精讲 【例1】 “走美”主试委员会为三?八年级准备决赛试题。 每 个年级12道题,并且至少有8道题与其他各年 级都不同。如果每道题出现在不同年级,最多只能出现 3次。本届活动至少要准备 道决赛 试题。 【考点】容斥原理之最值问题 【难度】4星 【题型】填空 【关键词】走美杯,4年级,决赛,第9题 【解析】每个年级都有自己8道题目,然后可以三至五年级共用 4道题目,六到八年级共用 4道题目,总共有 8 6 4 2 56 (道)题目。 【答案】56题 【例2】 将1?13这13个数字分别填入如图所示的由四个大小相同的圆分割成的 个圆内的7个数相加,最后把四个圆的和相加,问:和最大是多少? 【考点】容斥原理之最值问题 【难度】4星 【题型】填空 【解析】越是中间,被重复计算的越多,最中心的区域被重复计算四次,将数字按从大到小依次填写于 被重复计算多的区格中,最大和为: 13 X 4+ (12+11 + 10+9 ) X 3+ 8+7+6+5 ) X 2+ 4+3+2+1 ) =240. 【答案】240 【例3】如图,5条同样长的线段拼成了一个五角星?如果每条线段上恰有 这个五角星上红色点最少有多少个 ? 目 tMlF 13个区域中,然后把每 1994个点被染成红色,那么在

正弦扫频信号幅值及相位的提取

正弦扫频信号幅值及相位的提取 正弦振动控制系统提供输入的扫频信号,对于对数扫频, ,其中Sr为对数扫描率,若频响函数为则系统输出为 。 测量系统中可得到Calo信号及响应信号,通过对二者进行数据处理,可得到频域下的响应。不知道LMS的信号采集软件是如何提取频域响应的,个人认为软件计算速度有限,LMS应该是通过硬件实现的。下面我提供几种方法并进行比较。 算例对于Calo信号,频响函数为正弦扫频信号幅值及相位的提取,其中 ,信号采样率为1000次/秒,图1给出了时域下的响应信号。 图1时域下的响应信号 方法1 分段FFT 在[f, f+df]区间内对Calo信号、响应信号进行FFT变换,二者在频率f处的谱值比即为频响函数在f处的值。此方法的缺陷是由于信号采样率为1000Hz,而[f, f+df]的区间很窄,在此区间下时域的点不会很多,因而FFT的频率分辨率不高。 对于没有相位差的扫频信号,此方法能较好的提取幅值。图2给出了使用此方法提取的

幅值与理论结果比较,由图中可以看出二者基本吻合。 图2使用分段FFT提取的频域幅值 对于有相位差的扫频信号,则要对结果进行光滑处理,Matlab的smooth函数提供了这一功能。图3给出了有相位差时分段FFT提取的幅值与相位同理论结果的比较,从图中可以看出在频域峰值处分段FFT比理论值大,在其余频段二者吻合较好。

图3使用分段FFT提取的频域幅值、相位 分段FFT提取方法计算速度一般,不会出现异常而中止,计算精度基本也能保证。 方法2分段曲线拟合 在[f, f+df]区间内,假定A,ψ不变,此区间内在时域内对其拟合。图4给出了有相位差时曲线拟合提取的幅值与相位同理论结果的比较,从图中可以看出计算结果与真实值吻合非常好。 图 4 使用分段曲线拟合提取的频域幅值、相位 分段曲线拟合提取的结果精度非常高,但是由于是拟合方法,因而可能会由于初始值给的不合理或拟合关系式不恰当而出现迭代次数超过规定值从而导致计算中止。 由于相隔此次的频率相距很近,因而把上一次拟合的结果作为本次的初值,不但可以保证初始值给得非常合理,同时可以加快计算速度。另外要强调的是尽管如此,由于每个频率段都要使用拟合,因而分段曲线拟合方法计算速度比较慢。 方法3分段两点求解 在[f, f+df]区间内,利用两点求出两个未知数A,ψ,在[f, f+df]区间内对A,ψ取平均。图5给出了有相位差时曲线拟合提取的幅值与相位同理论结果的比较,从图中可以看出计算结果与真实值基本重合。由于算例中的扫频信号是理想的正弦扫频信号,因而两点求解能够精确计算得到真实值。

最值问题(费马点)

最值问题2 (费马点) 2、已知:P是边长为1的等边三角形ABC内的一点, PA+PB+PC的最小值.求PA+PB+PC的最小值.

3、(延庆)(本题满分 4分)阅读下面材料: 阅读下面材料: 1 ,在△ ABC (其中∠ BAC 是一个可以变化的角)小伟是这样思考的:利用变换和等边三角形将边的位置重新组合. 他的方法是以点 B 为旋转中心将△ ABP 逆时针旋转60。得到△ A BC,连接A 'A ,当点A 落在A C 上时,此 题可解(如图2). 请你回答:AP 的最大值是 ___________ . 参考小伟同学思考问题的方法,解决下列问题: 如图3,等腰Rt △ ABC .边AB=4,P ABC 内部一点, 则AP+BP+CP 的最小值是 ___________ .(结果可以不化简) 中, AB=2,AC=4 ,以BC 为边在 BC 的下方作等边△ PBC ,求AP 的最大值。 小伟遇到这样一个问题:如图 图1

4、(朝阳二模)阅读下列材料: 内部有一点P,连接FA、PB、PC,求PA+PB+PC的最小值. 小华是这样思考的:要解决这个问题,首先应想办法将这三条端点重合于一点的线段分离,然后再将它们连接成一条折线,并让折线的两个端点为定点,这样依据“两点之间,线段最短”,就可以求出这三条线段和的最小值了?他先后尝试了翻折、旋转、平移的方法,发现通过旋转可以解决这个问题?他的做法是,如图2,将厶APC绕点C顺时针旋转60o, 得到△ EDC ,连接PD、BE ,则BE的长即为所求. (1)请你写出图2中,PA+PB+PC的最小值为 __________ ; (2)参考小华的思考问题的方法,解决下列问题: ①如图3,菱形ABCD中,∠ ABC=60o,在菱形ABCD内部有一点P,请在图3 中画出并 指明长度等于PA+PB+PC最小值的线段(保留画图痕迹,画出一条即可); ②若①中菱形ABCD的边长为4,请直接写出当PA+PB + PC值最小时PB的长. 小华遇到这样一个问题,如图1, △ ABC 中,∠ ACB=30o, BC=6,AC=5,在^ABC 图1 图2 图3

2020年部编版小学奥数容斥原理之最值问题

小学奥数容斥原理之最值问题

1. 了解容斥原理二量重叠和三量重叠的内容; 2. 掌握容斥原理的在组合计数等各个方面的应用. 一、两量重叠问题 在一些计数问题中,经常遇到有关集合元素个数的计算.求两个集合并集的元素的个数,不能简单地把两个集合的元素个数相加,而要从两个集合个数之和中减去重复计算的元素个数,即减去交集的元素个数,用式子可表示成:A B A B A B =+-(其中符号“”读作“并”,相当于中文“和”或者“或”的意思;符号“”读作“交”,相当于中文“且”的意思.)则称这一公式为包含与排除原理,简称容斥原理.图示如下:A 表示小圆部分,B 表示大圆部分,C 表示大圆与小圆的公共部分,记为:A B ,即阴影面积.图示如下:A 表示小圆部分,B 表示大圆部分,C 表示大圆与小圆的公共部分,记为:A B ,即阴影面积. 包含与排除原理告诉我们,要计算两个集合A B 、的并集A B 的元素的个数,可分以下两步进行: 第一步:分别计算集合A B 、的元素个数,然后加起来,即先求A B +(意思是把A B 、的一切元素都“包含” 进来,加在一起); 第二步:从上面的和中减去交集的元素个数,即减去C A B =(意思是“排除”了重复计算的元素个数). 二、三量重叠问题 A 类、 B 类与 C 类元素个数的总和A =类元素的个数B +类元素个数C +类元素个数-既是A 类又是B 类的元素个数-既是B 类又是C 类的元素个数-既是A 类又是C 类的元素个数+同时是A 类、B 类、C 类的元素个数.用符号表示为:A B C A B C A B B C A C A B C =++---+.图示如下: 在解答有关包含排除问题时,我们常常利用圆圈图(韦恩图)来帮助分析思考. 知识要点 教学目标 7-7-5.容斥原理之最值问题 1.先包含——A B + 重叠部分A B 计算了2次,多加了1次; 2.再排除——A B A B +- 把多加了1次的重叠部分A B 减去. 图中小圆表示A 的元素的个数,中圆表示B 的元素的个数, 大圆表示C 的元素的个数. 1.先包含:A B C ++ 重叠部分A B 、B C 、C A 重叠了2次,多加了1次. 2.再排除:A B C A B B C A C ++--- 重叠部分A B C 重叠了3次,但是在进行A B C ++- A B B C A C --计算时都被减掉了. 3.再包含:A B C A B B C A C A B C ++---+.

基于新型相位幅值控制的三相PWM整流器数学模型(精)

第23卷第7期 2003年7月文章编号 中国电机工程学报 Proceedings of the CSEE TM461 文献标识码 Vol.23 No.7 Jul. 2003 ?2003 Chin.Soc.for Elec.Eng. 470?4051 基于新型相位幅值控制的三相PWM 整流器数学模型 张纯江1,2, 顾和荣1, 王宝诚1, 朱艳萍1, 刘彦民 oó±± 秦皇岛 066004A novel algorithm of the phase and amplitude control of three-phase source PWM rectifiers is presented. Thephase angle ξ of the front end voltage of a rectifier is regardedas the input variable and the output DC voltage UDC is used asthe output variable. The three-order transfer function betweenthe output DC voltage UDC and the phase angle ξ is deducedthrough the coordinate transforming and the small signalanalysis and is reasonably reduced to an one-order transferfunction. The internal relations of the system are unveiled. Themathematical model is used for the system design and thecontrol algorithm are implemented in a digital-signal processor.The theoretical analysis is verified by the simulations andexperiments. KEY WORDS: Mathematical model; PWM rectifier;Phase and amplitude; Power electroncs摘要 结合三相PWM 整流器的状态空间方程 并合理简化成为一阶传递函 数 为系统设计提供了理论依据 关键词脉冲宽度调制整流器电 力电子 1引言 系统数学模型尤其是以传递函数形式表示的数学模型是深入分析和设计系统的依据 国家自然科学基金项目

组合最值与操作问题

组合最值与操作问题 一.知识与方法 在数学竞赛中,经常有一些与组合问题相关的整最值问题,简称组合最值。所谓组合最值,就是指以整数、集合、点、线、圆等离散对象为背景,求它们满足某些约束条件的极大值或极小值。这类问题的解法与一般函数(连续变量)极值的解法有很大的差异。对于这类非常规的极值问题,要针对具体问题,认真分析,细心观察,选用灵活的策略与方法。通常可以从论证与构造两方面予以考虑。先论证或求得该变量的上界或下界,然后构造一个实例说明此上界或下界可以达到,这样便求得了该组合量的极大值或极小值。在论证或求解组合量的上界或下界时,通常要对组合量做出估计,在估计的过程中,构造法、分类讨论法、数学归纳法、反证法、极端原理、抽屉原理等起着重要的作用。 在数学竞赛中,还有一类操作问题。这类问题是指在一定的规则下,对给定的对象进行调整,探求被调整对象的初始状态或终止状态及其变化规律。 二.范例选讲 例1. m个互不相同的正偶数和n个互不相同的正奇数的总和为1987,对于所有这样的m与n,问3m+4n的最大值是多少?请证明你的结论。(1987年第二届全国数学冬令营试题) 思路分析:先根据题设条件求得3m+4n的一个上界,然后举例说明此上界可以达到,从而得到3m+4n的最大值。

解:设a 1,a 2,…,a m 是互不相同的正偶数,b 1,b 2,…,b n 是互不相同的正奇数,使得a 1+a 2+…+a m +b 1+b 2+…+b n =1987 (1) 这时分别有:a 1+a 2+…+a m ≥2+4+…+2m=m(m+1) (2) b 1+b 2+…+b n ≥1+3+…+(2n-1)=n 2 (3) 由①,②,③得m 2+m+n 2≤1987,因而有(m+2 1)2+n 2≤119874 + ④ 由④及柯西不等式,得3(m+2 1)+4n ≤ 4 119875)21(.432222+≤++ +n m , 由于3m+4n 为整数,所以3m+4n 221≤,⑤ 另一方面,当m=27,n=35时,m 2+m+n 2=1981<1987,且3m+4n=221。 故3m+4n 的最大值为221。 评注:在论证过程中用到了柯西不等式与一般二元一次不定方程的求解方法。 例2.设空间中有2n(n ≥2)个点,其中任何四点都不共面。将它们之间任意连接N 条线段,这些线段都至少构成一个三角形,求N 的最小值。 思路分析:通过构造实例,说明N ≥n 2+1,进而证明当N=n 2+1时,若在2n 点间连有N 条线段,则这些线段至少构成一个三角形。其证明过程可用数学归纳法或反证法或极端原理,在证明过程中要精打细算。 解法一:将2n 个已知点均分为S 和T 两组: S={A 1,A 2,…,A n },T={B 1,B 2,…B n }。 现将每对点A i 和B j 之间都连结一条线段A i B j ,而同组的任何两点

组合极值问题

组合极值问题 组合极值问题是各类数学竞赛的热点,它与代数,几何,数论等相比风格迥异。解组合极值问题往往需要某种技巧,因此,需要解题者具有丰富的解题经验与良好的题感。 1 构造法 我们常常通过构造抽屉,映射,表格等解决组合极值问题,有时还需要构造例子说明取到极值。 1.1构造抽屉 例1:(2000年中国数学奥林匹克)某次考试有5道选择题,每题都有4个不同答案供选择,每人每题恰好选1个答案。在2000份答案中发现存在一个 n ,使得任何 n 份答卷中都存在4份,其中每2份的答案都至多3题相同,求 n 的最小可能值。 解:将每道题的4种答案分别记为1 ,2 ,3 ,4 ,每份试卷上的答案记 为()k ,j ,i ,h ,g ,其中{ }4321,,,k ,j ,i ,h ,g ∈。令 ()()()(){}k ,j ,i ,h ,,k ,j ,i ,h ,,k ,j ,i ,h ,,k ,j ,i ,h ,4321,{}4321,,,k ,j ,i ,h ∈,共得256个四元组。 由于2000=256?7+208,故由抽屉原理知,有8份试卷上的答案属于同一个 四元组。取出这8份试卷后,余下的1992份试卷中仍有8份试卷属于同一个四元组,再取出这8份试卷,余下的1984份试卷中又有8份属于同一个四元组。又取出这8份试卷,三次共取出24份试卷。在这24份试卷中,任何4份中总有2份的答案属于同一个四元组,当然不满足题目的要求,所以 n ≥ 25 。 下面证明 n 可以取到25。令 ()(){}{}432140,,,k ,j ,i ,h ,g ,mod k j i h g |k ,j ,i ,h ,g S ∈≡++++=,则 |S| =256 ,且 S 中任何2种答案都至多有3题相同。从 S 中去掉6个元素,当余下的250种答案中的每种答案都恰有8人选用时,总有4份不相同。由于它们都在S 中,当然满足题目要求,这表明,n = 25满足题目要求。综上可知,所求的n 的最小可能值为25。 1.2构造映射 例2将正整数n 表示为一些正整数p a a a ,,,21Λ的和,即 p a a a n +++=Λ21,其中p a a a ≤≤≤Λ21,记)(n f 是如此表示的方法种数(如 4=4,4=1+3,4=2+2,4=1+1+2,4=1+1+1+1,故5)4(=f ),证明:对任意 )]2()([2 1 )1(,1++≤+≥n f n f n f n . 分析:由于本题证明的是一个非严格不等式,则需要构造一个单射或满射来证之。 证明:此题实质上是要证

一条线段最值问题的研究

页脚内容 l P A 图1 C B A D E 图2 C A E B D F G 图3 C B A M E D 一条线段最值问题的研究 初中阶段线段最值问题涉及广泛,有一条线段的最值问题,两条及多条线段和的最小值问题,还有两条线段差的最大值问题等,这里主要讨论一条线段最值问题。 基本原理:一条线段由两个点组成,不是动点即为定点,可以按照单动点和双动点分类,也可以按照动点形成的轨迹去分类。初中阶段考察最多的轨迹是直线和圆(或其部分),经过排列组合可以得到多种情形(这里不穷举,只讨论常见部分)。由于压轴题中动点形成的轨迹难以确定, 或者需要转化,所以也是造成部分优生答题困难的 重要原因。 基本情形1:一个定点和一个轨迹为定直线的 动点 如图,给定定点A和直线l上的动点P,利用 “垂线段最短”可知AP与l垂直时AP值最小。 例1.如图1,在⊿ABC中,∠CAB=30°,∠CBA=90°, BC=1,D为直线AB上一动点,把点D绕点C顺时针旋转 60°得点E,求BE的最小值. 分析:D的运动轨迹是直线,所以经过旋转后E的运 动轨迹也是直线,所以只需把直线AB绕点C顺时针旋转 60°可得E的轨迹,从而转化成点到直线的距离. 例2.如图2,在⊿ABC中,∠CAB=15°,AC=3,D为直线 AB上一动点(不与A、B重合), ⊿AED为等腰直角三角形且 ∠DAE=90°,过E作EF⊥DE,F为垂线上任一动点,G为DF的 中点,求线段CG的最小值. 分析:连接EG、AG 可得⊿ADG≌⊿AEG,从而可知∠ CAG=60°,所以G在直线上运动, 从而转化成点到直线的距 离. 例3.如图3,在直角⊿ABC中,CB=3,CA=4,M为斜边AB 上一动点,过M作MD⊥AC于D,过M作ME⊥BC于点E,求 线段DE的最小值为 . 分析:连接CM,则CM=DE,则只需求CM的最小值,从而双 动点转化成单动点,并且又一次转化成点到直线的距离. 例4.如图4,⊿ABC中,∠BAC=60°,∠ABC=45°,AB=2,D 是线段BC上的一个动点,以AD为直径画⊙O分别交AB,AC于E, F,连接EF,则线段EF长度的最小值为_________ .

中考压轴题突破:几何最值问题大全

中考压轴题突破:几何最值问题大全(将军饮马、造桥选址、胡不归、阿波罗尼斯圆等) 一、基本图形 所有问题的老祖宗只有两个:①[定点到定点]:两点之间,线段最短;②[定点到定线]:点线之间,垂线段最短。 由此派生:③[定点到定点]:三角形两边之和大于第三边;④[定线到定线]:平行线之间,垂线段最短;⑤[定点到定圆]:点圆之间,点心线截距最短(长);⑥[定线到定圆]:线圆之间,心垂线截距最短;⑦[定圆到定圆]:圆圆之间,连心线截距最短(长)。 余不赘述,下面仅举一例证明:[定点到定圆]:点圆之间,点心线截距最短(长)。 已知⊙O半径为r,AO=d,P是⊙O上一点,求AP的最大值和最小值。 证明:由“两点之间,线段最短”得AP≤AO+PO,AO≤AP+PO,得d-r≤AP ≤d+r,AP最小时点P在B处,最大时点P在C处。即过圆心和定点的直线截得的线段AB、AC分别最小、最大值。(可用“三角形两边之和大于第三边”,其实质也是由“两点之间,线段最短”推得)。 上面几种是解决相关问题的基本图形,所有的几何最值问题都是转化成上述基本图形解决的。 二、考试中出现的问题都是在基本图形的基础上进行变式,如圆与线这些图形不是直接给出,而是以符合一定条件的动点的形式确定的;再如过定点的直线与动点所在路径不相交而需要进行变换的。类型分三种情况:(1)直接包含基本图形;(2)动点路径待确定;(3)动线(定点)位置需变换。 (一)直接包含基本图形

例1.在⊙O中,圆的半径为6,∠B=30°,AC是⊙O的切线,则CD的最小值是。 简析:由∠B=30°知弧AD一定,所以D是定点,C是直线AC上的动点,即为求定点D到定线AC的最短路径,求得当CD⊥AC时最短为3。 (二)动点路径待确定 例2.,如图,在△ABC中,∠ACB=90°,AB=5,BC=3,P是AB边上的动点(不与点B重合),将△BCP沿CP所在的直线翻折,得到△B′CP,连接B′A,则B′A长度的最小值是。 简析:A是定点,B'是动点,但题中未明确告知B'点的运动路径,所以需先确定B'点运动路径是什么图形,一般有直线与圆两类。此题中B'的路径是以C为圆心,BC为半径的圆弧,从而转化为定点到定圆的最短路径为AC-B'C=1。 例3.在△ABC中,AB=AC=5,cos∠ABC=3/5,将△ABC绕点C顺时针旋转,得到△A'B'C,点E是BC上的中点,点F为线段AB上的动点,在△A'B'C 绕点C顺时针旋转过程中,点F的对应点是F',求线段EF'长度的最大值与最小值的差。 简析:E是定点,F'是动点,要确定F'点的运动路径。先确定线段A'B'的运动轨迹是圆环,外圆半径为BC,内圆半径为AB边上的高,F'是A'B'上任意一点,因此F'的运动轨迹是圆环内的任意一点,由此转化为点E到圆环的最短和最长路径。 E到圆环的最短距离为EF 2=CF 2 -CE==,E到圆环的最长距离为 EF 1=EC+CF 1 =3+6=9,其差为。 (三)动线(定点)位置需变换 线段变换的方法:(1)等值变换:翻折、平移;(2)比例变换:三角、相似。

初中数学竞赛专题复习第四篇组合第26章离散量的最大值和最小值问题试题新人教版

第26章 离散量的最大值和最小值问题 26.1.1** 某个篮球运动员共参加了10场比赛,他在第6、第7、第8、第9场比赛中分别得了23、14、11和20分,他的前9场比赛的平均分比前5场比赛的平均分要高,如果他的10场比赛的平均分超过18分,问:他在第10场比赛中至少得了多少分? 解析 设前5场比赛的平均得分为x ,则前9场比赛的平均得分为 523141120568 99x x +++++= . 由题设知 568 9 x x +>, 解得17x <.所以前5场最多得分是 517184?-=(分). 再设他第10场比赛得了y 分,那么有 84681810180y ++>?=, 解得28y >y>28. 故他第10场比赛得分≥29分. 另一方面,当他在第6、第7、第8、第9、第10场比赛中分别得了23、14、11、20和29分,前5场总得分为84分时,满足题意. 所以,他在第10场比赛中至少得了29分. 评注 在解最大值(或者最小值)问题时,我们常常先估计上界(对于最小值,估计下界),然后再构造一个例子说明这个上界(或者下界)是能够取到的,只有这样,才完整地解决了问题. 26.1.2* 从任意n 个不同的正整数中,一定可以从中找到两个数,它们的差是12的倍数,求n 的最小值. 解析 任取13个不同的整数,它们除以12所得到的余数中,一定有两个相同,于是它们的差是12的倍数. 又l ,2,…,12这12个数,其中没有两个数的差为12的倍数. 综上所述,至少需任取13个数才能满足题意. 26.1.3** 从1,2,3,…,20中,至少任取多少个数,可使得其中一定有两个数,大的数是小的数的奇数倍. 解析 从1,2,…,20中取7,8,…,20这14个数,其中没有一个数是另一个数的奇数倍. 把1,2,…,20分成如下14组:{1,3,9},{2,6,18},{4,12},{5,15},{7},f8},{10},{11},{13},{14},{16},{17},{19},{20},从中任取15个数,一定有两数取自同一组,于是大数便是小数的奇数倍. 26.1.4** 如果甲的身高或体重至少有一项比乙大,则称甲不亚于乙;在100个小伙子中,如果某人不亚于其他99人,就称他为棒小伙子.问100个小伙子中的棒小伙子最多可能有多少个? 解析 取100个小伙子是这样的一种特殊情况.他们的身高互不相同,是从小到大排列的,他们的体重也互不相同,且是从大到小排列的,这样的100个小伙子都是棒小伙子,所以棒小伙子最多有100个. 26.1.5** 代数式rvz rwy suz swy tux tvx --++-中,r 、s 、t 、u 、v 、w 、x 、y 、z 可以分别取1或者1-. (1)求证:代数式的值都是偶数; (2)求该代数式所能取到的最大值. 解析 (1)因为 ()11111110mod2rvz rwy suz swy tux tvx --++-≡++++++≡, 所以,此代数式的值为偶数.

几何最值问题参考标准答案

几何最值问题参考答案

————————————————————————————————作者:————————————————————————————————日期:

几何最值问题 一.选择题(共6小题) 1.(2015?孝感一模)如图,已知等边△ABC的边长为6,点D为AC的中点,点E为BC 的中点,点P为BD上一点,则PE+PC的最小值为() A.3B.3C.2D.3 考点:轴对称-最短路线问题. 分析:由题意可知点A、点C关于BD对称,连接AE交BD于点P,由对称的性质可得,PA=PC,故PE+PC=AE,由两点之间线段最短可知,AE即为PE+PC的最小值. 解答:解:∵△ABC是等边三角形,点D为AC的中点,点E为BC的中点,∴BD⊥AC,EC=3, 连接AE,线段AE的长即为PE+PC最小值, ∵点E是边BC的中点, ∴AE⊥BC, ∴AE===3, ∴PE+PC的最小值是3. 故选D. 点评:本题考查的是轴对称﹣最短路线问题,熟知等边三角形的性质是解答此题的关键. 2.(2014?鄂城区校级模拟)如图,在直角坐标系中有线段AB,AB=50cm,A、B到x轴的距离分别为10cm和40cm,B点到y轴的距离为30cm,现在在x轴、y轴上分别有动点P、Q,当四边形PABQ的周长最短时,则这个值为() A.50 B.50C.50﹣50 D.50+50 考点:轴对称-最短路线问题;坐标与图形性质.

专题:压轴题. 分析:过B点作BM⊥y轴交y轴于E点,截取EM=BE,过A点作AN⊥x轴交x轴于F点,截取NF=AF,连接MN交X,Y轴分别为P,Q点,此时四边形PABQ的周长最短,根据题目所给的条件可求出周长. 解答:解:过B点作BM⊥y轴交y轴于E点,截取EM=BE,过A点作AN⊥x轴交x轴于F点,截取NF=AF,连接MN交x,y轴分别为P,Q点, 过M点作MK⊥x轴,过N点作NK⊥y轴,两线交于K点. MK=40+10=50, 作BL⊥x轴交KN于L点,过A点作AS⊥BP交BP于S点. ∵LN=AS==40. ∴KN=60+40=100. ∴MN==50. ∵MN=MQ+QP+PN=BQ+QP+AP=50. ∴四边形PABQ的周长=50+50. 故选D. 点评:本题考查轴对称﹣最短路线问题以及坐标和图形的性质,本题关键是找到何时四边形的周长最短,以及构造直角三角形,求出周长. 3.(2014秋?贵港期末)如图,AB⊥BC,AD⊥DC,∠BAD=110°,在BC、CD上分别找一点M、N,当△AMN周长最小时,∠MAN的度数为() A.30°B.40°C.50°D.60° 考点:轴对称-最短路线问题. 分析:根据要使△AMN的周长最小,即利用点的对称,使三角形的三边在同一直线上,作出A关于BC和CD的对称点A′,A″,即可得出∠AA′M+∠A″=∠HAA′=70°,进而得出∠MAB+∠NAD=70°,即可得出答案.

相关文档
最新文档