Splines

Cubic Hermit Interpolation Polynomial

Given a list of points and the first derivative at each point construct a cubic spline curve through the points. This example is in 2D Cartesian coordinate system. p are not vectors but scalars.

It is not too hard to see how this could get complicated. For example consider generalizing to 3D where the boundary is not a point but a curve. Now consider a generalized triangle which has three boundaries which instead of being straight lines are curves. Find a surface that is constrained by the boundaries (imagine a mesh of these generalized triangles then how are they to be connected in a continuous way). Then extend the problem so that the surface is continuous in its first derivative.

In contrast the Cubic Hermite Interpolation in Vector Space is a easy way to fit a curve between two points in any dimension. I will have to start reading for a similar successful solution for generalized triangles in 3D.

Let
$p_{0} = p (0)$
$p_{1} = p (1)$
$p'_{0} = p' (0)$
$p'_{1} = p' (1)$
t varies between 0 and 1.

$p (t) = c_{0} + c_{1} t + c_{2} t^{2} + c_{3} t^{3}$

$t = 0$ then
$c_{0} = p_{0}$

$p (t) = p_{0} + c_{1} t + c_{2} t^{2} + c_{3} t^{3}$
$p' (t) = c_{1} + 2 c_{2} t + 3 c_{3} t^{2}$
$t = 0$ then
$c_{1} = p'_{0}$

$t = 1$ then
$p_{1} = p_{0} + p'_{0} + c_{2} + c_{3}$
$p'_{1} = p'_{0} + 2 c_{2} + 3 c_{3}$

Solving these two simultaneous equations.

$p (t) = p_{0} + p'_{0} t + (3 p_{1} - 3 p_{0} - 2 p'_{0} - p'_{1}) t^{2} + (p'_{1} + p'_{0} + 2 p_{0} - 2 p_{1}) t^{3}$

Continuous Surface from Curves

Consider two curves in 3D space. Let both curves be parameterized in t from 0 to 1 to travel from the start of the curve to the finish.

To construct a surface between the two points use the vector space interpolation between two corresponding points on the other curve.

This is a generalized rectangle where two of the boundaries have fixed curves and the other two are results of the interpolation.

Note that the fixed curve itself could have been generated from sample points and be built with splines.

While two sides are continuous the other two sides are not. When these have to be sown in the mesh could so easily have holes. I am just beginning to look at these issues.

Cubic Interpolation with Barycentric Coordinates

This is something I cooked up after looking at Cubic Hermite Interpolation in Vector Space and reading about Barycentric Coordinates.

First have a cubic multiplied by the point or its derivative. As the variable t goes to zero the approximation should not contribute then f(0)=0 and f'(0)=0 are the natural boundary conditions.

$f (0) = 0$
$f' (0) = 0$
$f = a t^{3} + b t^{2}$

For the point p_i it should be zero when the derivative is taken else 1.

$f (1) = 1$
$f' (1) = 0$
$f = - 2 t^{2} + 3 t^{3}$

For the point p'_i it should be one when the derivative is taken else zero.

$f (1) = 0$
$f' (1) = 1$
$f = t^{3} - t^{2}$

With Barycentric coordinates it becomes the following interpolation.

$0 \leq u_{i} \leq 1$
$\sum u_{i} = 1$
$p = \sum (- 2 u_{i}^{3} + 3 u_{i}^{2}) p_{i} + (u_{i}^{3} - u_{i}^{2}) p'_{i}$

Note that how this formula was built is similar to how the cubic interpolation formula was built. The main difference is that the cubic interpolation had point t=0 as a real point where I use it as a zero. I am also sure that this is a known result (if it works, I have not tested it yet).

Motivation for this comes from wanting to interpolate a simplex with a smooth surface. Also I am interested in the equivalent of taylor series in 3D.

This can be extended to a second order approximation with degree 5 having the boundary conditions f(0) = f'(0) = f''(0) = 0 and the others specific to the point. For example f(1)=f'(1)=0 and f''(1)=1 with f*p''₀.

$p = \sum a_{i} (u_{i}) p_{i} + b_{i} (u_{i}) p'_{i} + c_{i} (u_{i}) p''_{i}$

Two Point Interpolation Generalized

The idea is very simple. First let the first K derivatives at the two points be known. Then given this information let each point be summed uniquely to give its own contribution to the interpolation function f. This must incorperate knowledge of the derivatives.

Let $p^{(i)}$ be the first k derivatives of point.
$i = 0 ... k - 1$ where $i = 0$ is the point in space.

To fit k condtions requires k variables.

$f (t) = \sum_{i = 0}^{k - 1} c_{i} t^{i}$
$p^{(i)} = f (0)^{(i)}$

Now consider the effect of summing each of the two points. They would interfere with eachother. So to cancel out the interference for the second point, we first have to say where the second point is. Naturally the second point is when $t = 1$ . This doubles the number of conditions that need to be made. So reformatting the problem.

$f_{0} (t) = \sum_{i = 0}^{2 k - 1} c_{0 i} t^{i}$
$p_{0}^{(i)} = f_{0} (0)^{(i)}$
$0 = f_{0} (1)^{(i)}$

Similarly for the second point $p_{1}$ .

$f_{1} (t) = \sum_{i = 0}^{2 k - 1} c_{1 i} t^{i}$
$p_{1}^{(i)} = f_{1} (1)^{(i)}$
$0 = f_{1} (0)^{(i)}$

Curve Boundaries on 3D Triangle

The idea is simple. Use symmetric splines on the boundaries of a triangle so that as the point moves on the surface towards one of these curves it converges to it.

When on one of these boundaries the other point is non-existant. Hence the mesh can be stiched so that two neighboring surfaces connect.

Choose the boundary curve to be a cubic polynomial such that the first derivative is also satisfied at the points. So that the points of the triangle have C¹ continuity but the intersection curves on the boundary have C⁰ continuity.

While this may at first appear to be a compromise it is strategic. For problems that I have the points are the ultimate sources of knowledge. What is between the points is unknown.

Ofcourse the next stage would be to get C¹ at the boundary as well, I am really going to have to speak with someone who knows surfaces and their algebraic representations. But what will probably happen is that I spend the next year teaching myself. Rational bicubic B-spline surfaces possess C² however they do not satisfy the boundary conditions at the points, if they can be made to do this and their boudary edges are also C² and the continuity could be generalized for any dimension I would be very happy (if I could do it).

$k = i + 1 \mod 3$
$f_{i} = A_{i} P_{i} + B_{i} P'_{i} + C_{i} P_{k} + D_{i} P'_{k}$
$f_{i} (u_{i})$ as
$A_{i} (u_{i})$ , $B_{i} (u_{i})$ , $C_{i} (u_{i})$ , $D_{i} (u_{i})$ .

A point on the surface can be expresses as a sum of the functions.

$p = \sum_{i = 0}^{2} f_{i} w_{i}$
where $w_{i}$ is the weight.

$w_{i} = \frac{v_{i}^{- 1}}{v_{0}^{- 1} + v_{1}^{- 1} + v_{2}^{- 1}}$

Let $u_{i}$ be a point on the boundary.
Let $v_{i}$ be the minimum distance from a point $q$ to $u_{i}$ where $q$ is some point inside the triangle formed by $P_{i}$ .

Essentially the problem is given point q find point p. q exists in the triangle. Let q be the orthocenter and calculate v_i. u_i is the ratio of the two points on the boundary and since this function is symmetric u_i is easily calculated. So together u_i and v_i form a local coordinate system.

References

Tomas Akenine-Moller and Eric Haines (2002). Real-Time Rendering 2nd Edition. ISBN 1-56881-182-9