Splines

Question 1

The idea behind curve segments

Answer 1

An editable representation of a curve with “nice” properties (maintains its shape as a curve after editing)

Question 2

Linear curve segments

Answer 2

An nth degree curve segment is created from the affine combination of (n-1)-degree segments.

Eg. Line segment is an affine combination of points.

Question 3

deCastelajau Algorithm

Answer 3

Find a point on a curve segment.
1. Join points Pi by line segments
2. While # segments > 1, Join the t: (1-t) points of those line segments by line segments
3. The interpolated point on the final line segment is a point on the curve
And the final line segment is tangent to the curve at t

Question 4

2 Properties of Bernstein Polynomials

Answer 4

1. Partition of unity: the sum of bernstein polynomials of degree n = 1
2. Non-negativity: bernstein bases are all positive.

Question 5

Bernstein Polynomials

Answer 5

A basis space for representing polynomials based on affine combinations of linear curve segments. Can be evaluated recursively

Question 6

Hermite Interpolation

Answer 6

Only define the tangent vector at starting point vertices.

Question 7

Piecewise polynomials

Answer 7

Have continuity of 1st and 2nd derivatives

Question 8

Catmull-Rom Splines

Answer 8

Just specify the points and cubic hermite will make up the derivatives from the points. Tangent at point i is equal the slope between points i-1, i+1.

Question 9

B-Splines

Answer 9

Splines with local manipulation and piecewise definition. Must create our own basis. Can use bernstein polynomials.

Question 10

Uniform B-Spline

Answer 10

When all knots on the spline are evenly spaced

Question 11

Bezier Patches

Answer 11

Use the same bezier spline techniques to control a polygon. Use patch basis functions.