15.7 bicubic bezier surface patches - computer science ...cheng/cs535/notes/cs535-curves-3.pdf ·...
TRANSCRIPT
1
15.7 Bicubic Bezier Surface Patches
CS Dept, UK
where
2
15.7 Bicubic Bezier Surface Patches
CS Dept, UK
Matrix form:
3
15.7 Bicubic Bezier Surface Patches
CS Dept, UK
Satisfies convex hull property
Subdivision process
- Subdivide in u and then subdivide in v
Rendering techniques
-Wire frame: generate iso-parametric curves in both directions
-Shaded images:
Ray tracing
Scan convert approximating polygons:
approximate the surface patch by fine polygons
(triangles or quadriterals), then shade the polygons
Patches can be joined to form complicated shapes
4
15.8 Subdivision Techniques for
Piecewise Surfaces
CS Dept, UK
Midpoint subdivision
(see previous section)
5
15.9 Non-Uniform B-Spline Curves
CS Dept, UK
6
15.9 Non-Uniform B-Spline Curves
CS Dept, UK
Intuitively, B-splines of degree n ( order n +1) are piecewise
polynomial curves that are zero at all subintervals but n +1
of them and have continuous (n -1)st derivative. The following
are examples of B-splines of degree 0, 1, 2, and 3.
7
15.9 Non-Uniform B-Spline Curves
CS Dept, UK
8
15.9 Non-Uniform B-Spline Curves
CS Dept, UK
9 CS Dept, UK
15.9 Non-Uniform B-Spline Curves
10 CS Dept, UK
15.9 Non-Uniform B-Spline Curves
11 CS Dept, UK
15.9 Non-Uniform B-Spline Curves
12 CS Dept, UK
15.9 Non-Uniform B-Spline Curves
13 CS Dept, UK
15.9 Non-Uniform B-Spline Curves
14 CS Dept, UK
What are the relationship between the uniform cubic B-
splines defined here and the cubic B-spline blending
functions defined in Section 15.3?
15.9 Non-Uniform B-Spline Curves
15
15.9 Non-Uniform B-Spline Curves
CS Dept, UK
16
15.9 Non-Uniform B-Spline Curves
CS Dept, UK
17
15.9 Non-Uniform B-Spline Curves
CS Dept, UK
Example of a cubic B-spline curve:
If knot for
all , then we
get a uniform
cubic B-spline
curve. In that
case, would the
curve defined
here be the same
as the one given
in Section 3.1.3?
iti i
18
15.9 Non-Uniform B-Spline Curves
CS Dept, UK
19
15.9 Non-Uniform B-Spline Curves
CS Dept, UK
20
15.9 Non-Uniform B-Spline Curves
CS Dept, UK
21
15.9 Non-Uniform B-Spline Curves
CS Dept, UK
22
15.9 Non-Uniform B-Spline Curves
CS Dept, UK
23
15.9 Non-Uniform B-Spline Curves
CS Dept, UK
Answer to Question 1: The above theorem shows that a cubic B-spline curve
satisfies a stronger convex hull property: each segment
of a (non-uniform) cubic B-spline curve is contained in
the convex hull of the four control points that determine
the segment.
Answer to Question 2: The resulting cubic B-spline curve interpolates the first
and last control points.
Why?
24
15.9 Non-Uniform B-Spline Curves
CS Dept, UK
25
15.9 Non-Uniform B-Spline Curves
CS Dept, UK
26
15.9 Non-Uniform B-Spline Curves
CS Dept, UK
27
15.9 Non-Uniform B-Spline Curves
CS Dept, UK
28
15.9 Non-Uniform B-Spline Curves
CS Dept, UK
This chart will
be used in
the solution
for question 3.
29
15.9 Non-Uniform B-Spline Curves
CS Dept, UK
This chart will
be used in
the solution
for question 3.
30
15.9 Non-Uniform B-Spline Curves
CS Dept, UK
31
15.9 Non-Uniform B-Spline Curves
CS Dept, UK
),[ 43 ttt
32
15.9 Non-Uniform B-Spline Curves
CS Dept, UK
33
15.9 Non-Uniform B-Spline Curves
CS Dept, UK
34
15.9 Non-Uniform B-Spline Curves
CS Dept, UK
35
15.9 Non-Uniform B-Spline Curves
CS Dept, UK
36
15.9 Non-Uniform B-Spline Curves
CS Dept, UK
Why?
37
15.9 Non-Uniform B-Spline Curves
CS Dept, UK
38
15.9 Non-Uniform B-Spline Curves
CS Dept, UK