1. This Week Week Topic Week 1 Week 2 Week 3 Week 4 Week 5
Coordinate Systems, Basic Functions
Week 2
Trigonometry and Vectors (Part 1)
Week 3
Vectors (Part 2)
Week 4
Vectors (Part 3: Locus)
Week 5
Tutorial A: Question and answer session for weeks 1-4
Week 6
Matrices (Part 1)
Week 7
Matrices (Part 2) and Transformations
Week 8
Complex numbers
Week 9
Curves
Week 10
Tutorial B: Question and answer session focusing on weeks 6-9

2. Short Course in: Mathematics and Analytic Geometry
Week 9 Curves

3. LERP – Linear Interpolation
Given two position vectors r1 and r2, a linear interpolation is a straight line joining their respective points P1 and P2.

4. LERP – Linear Interpolation
Oh! But, hang on a minute! LERP is just the line equation defined in the interval: 0  t  1:

5. SLERP – Spherical Linear Interpolation
SLERP is an extension on LERP to the case of a curve interpolation on a spheroid.
In the most simplest case, we have two orthogonal unit position vectors r0 and r1 and we interpolate between 0 and π/2 radians with a parameter in the interval 0  t  1.

6. SLERP – Spherical Linear Interpolation
In the general case, position vectors r0 and r1 are not orthogonal and we want to interpolate between 0 and some angle .

7. SLERP – Spherical Linear Interpolation
The general SLERP
Formula is derived as follows:

8. Quaternion SLERP
Instead of vectors we can plug in unit quaternion to interpolate unit quaternion at parametric angles:
Where:

9. Parametric Curves
Any 2D curve can be projected in 3D space onto a parametric plane:

10. Parametric Curves
Similarly, in polar coordinates where f() define radius length:
Or in cylindrical space (in this case, an elliptic spiral):

11. Bezier Curves
However, some parametric curves can be inflexible and difficult to design.
Ideally, we would like to construct curves in a predictable way from some fixed points.
We could join points with LERPs to form a piecewise curve:

12. Bezier Curves But, the curve is not smooth: Bezier curves can be
constructed as a recursive system of LERPs (De Casteljau's algorithm). For example:

13. Bezier Curves

14. Bezier Curves
The general form of Bezier curves follows a binomial expansion pattern:

15. Inverse Bezier Let us assume we want a cubic Bezier curve:
That interpolates points p1 and p2 at times t1 and t2 respectively (remember, the ends are fixed):
We solve for control points r1 and r2, by finding the inverse of the matrix:

16. Catmull-Rom Spline
Using Tangents like this, curves can be joined with C1 continuity.

17. Cubic B-Splines
With B-Splines, continuity is always one degree lower than the degree of each curve piece. Therefore, a cubic B-Spline has continuity C2. The following curve connects pi to pi+1, for 0t 1:
Cubic B-Splines do not interpolate their end points.

18. Cubic B-Splines
In general, a B-Spline can be defined as follows (the ti are called knots):

19. Bezier Surfaces
Bezier curves can be extended to surfaces on unit squares.

20. B-Spline Surfaces