Download B-Splines: Non-Uniform Curve Interpolation with Basis Functions and Subdivision and more Study notes Computer Graphics in PDF only on Docsity!
B-Splines
Like Catmull–Rom splines, start with sequence of points
Curves no longer interpolate control points
- points where segments actually meet are called knots
- for Hermite et al the knots were always control points
Lack of interpolation isn’t a big problem for interactive design
- but it’s hard to predict curve just based on points coordinates
p
0
,... , p
n
− 3
2 3 − 2
− 1
i
i
i
i
u u u u
p
p
p
p
p
B-Spline Basis Functions
Non-negative functions
- implies convex hull property
b ( u )
1
b ( u )
4
b ( u )
3
b ( u )
2
( )
( )
( )
b u u
b u u u
b u u u u
b u u
3
1
3 2
2
3 2
3
3
4
i i i i
b u b u b u b u
1 − 3 2 − 2 3 − 1 4
p + p + p + p
Converting Between Cubic Spline Types
We saw a specific example of Bézier–Hermite conversion
Suppose we want to convert between two arbitrary splines
Given geometry matrix G
find equivalent G
for other spline
0 0
3 1
0 2
3 3
p p
p p
r p
r p
1 1 2 2
T T
u M G u M G
− 1
2 2 1 1
G = M M G
Drawing Spline Curves
Method #1 — Direct evaluation
- we have a function that generates points on the curve
- vary parameter u between 0 and 1
- substitute into formula and compute a position
- connect consecutive points with line segments
Method #1a — Direct evaluation with forward differencing
- instead of evaluating polynomials directly
- incrementalize polynomial to cut down on multiplies
This approach has some problems
- uniform parameter spacing is not uniform in space
- length of segments will vary over line
- control over length is important
- too long makes jagged curves; too short is too slow to draw
Modeling By Subdivision
Recall that we can draw spline curves via subdivision
- start with the control polyline
- recursively subdivide until “smooth enough”
- and draw the individual line segments
We can actually use this as a modeling primitive
- define the curve as limit of infinite number of subdivision steps
- throw out all our polynomials
Developing Subdivision Curves
Assume that we have some control polygon
- a closed piecewise-linear curve in the plane
Need two fundamental operations:
- Linear Subdivision — introduce new vertices
- Linear Smoothing — modify positions of vertices
Linear Subdivision of Curves
Split each edge of the curve at its barycenter (midpoint)
- thus doubling the number of vertices
v
i
v
i+ 1
(v
i
+ v
i+ 1
Linear Smoothing of Curves
Reposition each vertex at weighted combination of neighbors
Can also rewrite the above in a matrix form
i − 1
v
i
v
i + 1
v
i i i i
1 − 1 2 + 1
v = v + v + v
i
i
[ ]
i
i i
i
− 1
1 2 3
v
v v
v
Subdivision as Linear Operator
Points after k steps are linear combinations of previous points
- can therefore write subdivision step as a matrix operation
k k k
i i
k
i i
n n
n n
x y
x y
x y x y
x y
x y
x y
− 1 − 1
1 1
1 1
2 2 2 2
− 1
2 + 1 2 + 1
2 2
=
=
p S p
S
M M
M M
M M
Smoothing As Barycentric Averaging
**1. Compute barycenters of adjacent edges
- Compute barycenter of barycenters**
Same as weights [! " !] but works in higher dimensions too
i
′ v
i − 1
v
i
v
i + 1
v
