Total review

Sa
1
0 0 3 2
0 1 2 1
0 0 1 1
C=
0 3 2
Sa = [P0 P1 t0 t1]T Sb = [P1 P2 t1 t2]T
Applying C2 continuity:
ra’’(1) = 6P0 – 6 P1 + 2t0 + 4t1 rb’’(0) = - 6P1 + 6 P2 - 4t1 - 2t2
with 0
u
1
On evaluating the Bezier equation and its derivative at u = 0, 1
r(0) = V0 r(1) = Vn r’(0) = n (V1 – V0) r’ (1) = n (Vn – Vn-1)
Bezier Curve Model
C nj + i =
(n + i )! j !(n + i j )!
B-spline Model
n
Quadratic uniform B-spline model with control points V0, V1, and V2
1
r(t) =
2 2 1
1 0 0
V0 V1 V2
= U3 M3 P3
n
Cubic (n = 3) Bezier curve model r(u) = (1 – u)3 V0 + 3u(1 – u)2 V1 + 3u2(1 – u) V2 + u3 V3
1 3 6 3 0 3 1 3 0 0 0 0 0 V0 V1 V2 V3
r( u ) =
u3
u2
u 1
3 3 1
=UMR
n
Course exercises 5times (4marks each time)
n
n
course paper (20%)
Examination
nopen book paper (CAD 90% plus CAM 10%), Calculation problems(80%) and short answer questions(20%)
t1 t0 P0 r(u) P1
r (u) = U A = U MV with 0
2 = u3 u 2 u 1 3 0 1 2 3 0 0 1
u
1 2 1 0
1
P0 1 P1 0 t0 0 t1 2u 3 – 3u 2 + 1 P0 P1 2u3+3u 2 = HV t0 u3 – 2u 2 + u t1 u3 – u2
½
[t2 t 1]
2 1
0≤t≤1
§ Cubic uniform B-spline model with control points V0, V1, V2, and V3
1
r(t) = 1/6 [u3 u2 u 1]
3 6 0 4
3 3 1
3 1 V0 3 0 V1 3 1 0 V2 0 V3
= U4 M4 P4
An end tangent is estimated by fitting a circle through the 3 points at the end t0 = | a | (r c) / | r c | Where a = P1 – P0 ; b = P2 – P0 ; c = a b r = {|a|2 (b c) + |b|2 (c a)} / (2|c|2)
n
Cubic Spline Fitting
n
Estimation of end tangents, t0 and tn
n n n
Circular end condition Polynomial end condition Free end condition
Cubic Spline Fitting - Circular end condition
B-spline Model defined by n+1 points Vi is given by the
r (u ) =
n i=0
Vi N i,n (u )
with 0
u
1
1 n-i j j n ( 1) C (u+n-i-j) n +1 Ni,n(u) = n ! j= 0
Where
The primary function
P0 P1 P2 P3
In general, a degree-n polynomial curve can be used to fit (n + 1) data points.
r (u)
=
n
i = 0
a
i
u
i
Ferguson Curve Model
n
Constructing a curve segment: n Joining two end points P0 and P1; n Having specified end tangents t0 and t1 i.e., P0 = r (0); P1 = r (1); t0 = r’ (0); t1 = r’ (1)
Chapter 2: Curves
n
Four curve models
n n n n
Standard polynomial curve Ferguson curve Bezier curve B-spline curve
n
Curve fitting
Polynomial Curve Models
Curve Segment Definition: A cubic polynomial curve model: r (u) = a +bu +cu2 + du3 used in representing a curve segment is specified by its end conditions, e.g., (a) (b) 4 points (P0, P1, P2 and P3) or two end points P0 and P1; two end tangents t0 and t1.
V0
V1
V2
V3
n
B-spline curves and Bezier curves have many advantages in common n Control points influence curve segment shape in a predictable, natural way, making them good candidates for use in an interactive design environment. n Both types of curve are axis independent, multivalued, and both exhibit the convex hull property. B-spline curves have advantages over Bezier curves: n Local control of curve shape. n The ability to add control points without increasing the degree of the curve.
0 ≤ t≤ 1
Parametric Continuity Condition
t0
r a( u )
P0
P1
t2
rb(u)
P2
t1 = ?
Two curve segments ra(u) and rb(u) q q q q
ra(1) = P1 = rb(0) ra’(1) = t1 = rb’(0) ra’’(1) = rb’’(0)
Chapter 1: Instruction
n
n n n
What is CAD/CAM/CAE/CAPP? How is the relationship among them?) What is the HISTORY of CAD/CAM? Hardware and software of CAD/CAM system? What is Geometric Modelling and its typical applications?
n
n
ti-1 + 4ti + ti+1 = 3(Pi+1 – Pi-1)
for i = 1, 2, …, n – 1
B-spline Model
n
On evaluating the cubic B-spline (k = 4) and its derivative at t = 1, 0, r (0) = [4V1 + (V0 + V2)] / 6 r(1) = [4V2 + (V1+ V3)] /6 r’(0) = (V2 – V0) / 2 r’(1) = (V3 – V1) / 2
Cubic Spline Fitting (Ferguson Model)
§ Employing Ferguson curve model ra (u) = U C rb (u) = U C Sb with 0 u 1 U= [u3 u2 u 1]
C0-continuity and C1-continuity already applied
Cubic Spline Fitting（Ferguson Model)
n
Applying parametric C2-condition
t0 + 4t1 + t2 = 3(P2 – P0)