Collision with Ellipse - geometry

Question

For an elliptical billiard-table, how can a collision be detected and resolved between the boundary of this table and one billiard-ball?
1.) I want to see if the position, P(x,y), of the billiard-ball lies
inside
on
or outside the boundary of the ellipse. [update: part 1. is solved]
2.) If it lies on or outside the boundary, the new velocity has to be calculated (just flipping the velocity is not enough).
3.) If it lies outside, it has to be moved back to lie on the boundary first.
========
==== * ====
==== ====
= =
==== ====
==== ====
========
Given is the position P(x,y) and the velocity V(x,y) of the billiard ball, plus the position of the center of the ellipse C(x_0,y_0) and both semi-axes a,b of the ellipse.

Answer 1

Just use the equation of the ellipse like you used the equation for the circle:
((p.x-x0)/a)^2 + ((p.y-y0)/b)^2 = k
If k < 1 -> inside the ellipse
If k == 1 -> on the ellipse
If k > 1 -> outside the ellipse

Answer 2

Since you are considering an elliptical (thus convex) board, I suppose you could use something based on GJK. You would get the point of contact and a surface normal during a collision, and the minimum distance between the objects and the associated witness points if there is no collision.
Collision detection with GJK is very fast and you could implement it for other shapes very easily (you only need to recode the support function). For an ellipse, I think the support function would be something like this (try to verify that) :
h=((x^2)/(a^4)+(y^2)/(b^4))^(-1/2)
Some links :
the initial paper GJK

Answer 3

Some funny experiments with elliptic table. Delphi code (without error handling!).
//calculates next ball position in ellipse
//ellipse semiaxes A, B, A2 = A * A, B2 = B * B
//center CX, CY
//PX,PY - old position, VX,VY - velocity components
//V - scalar velocity V = Sqrt(VX * Vx + VY * VY)
procedure TForm1.Calc;
var
t: Double;
eqA, eqB, eqC, DD: Double;
EX, EY, DX, DY, FX, FY: Double;
begin
//new position
NPX := PX + VX;
NPY := PY + VY;
//if new position is outside
if (B2 * Sqr(NPX) + A2 * Sqr(NPY) >= A2 * B2) then begin
//find intersection point of the ray in parametric form and ellipse
eqA := B2 * VX * VX + A2 * VY * VY;
eqB := 2 * (B2 * PX * VX + A2 * PY * VY);
eqC := -A2 * B2 + B2 * PX * PX + A2 * PY * PY;
DD := eqB * eqB - 4 * eqA * eqC;
DD := Sqrt(DD);
//we need only one bigger root
t := 0.5 * (DD - eqB) / eqA;
//intersection point
EX := PX + t * VX;
EY := PY + t * VY;
//mark intersection position by little circle
Canvas.Ellipse(Round(EX - 2 + CX), Round(EY - 2 + CY),
Round(EX + 3 + CX), Round(EY + 3 + CY));
//ellipse normal direction
DX := B2 * EX;
DY := A2 * EY;
DD := 1.0 / (DY * DY + DX * DX);
//helper point, projection onto the normal
FX := DD * (NPX * DX * DX + EX * DY * DY - DY * DX * EY + DX * DY * NPY);
FY := DD * (-DX * DY * EX + DX * DX * EY + DX * NPX * DY + DY * DY * NPY);
//mirrored point
NPX := NPX + 2 * (EX - FX);
NPY := NPY + 2 * (EY - FY);
//new velocity components
DD := V / Hypot(NPX - EX, NPY - EY);
VX := (NPX - EX) * DD;
VY := (NPY - EY) * DD;
end;
//new position
PX := NPX;
PY := NPY;
//mark new position
Canvas.Ellipse(Round(PX - 1 + CX), Round(PY - 1 + CY),
Round(PX + 1 + CX), Round(PY + 1 + CY));
end;
A = 125, B = 100
Start from ellipse center (left picture) and from the right focus point (right picture), ball reaches left focus, then returns to right focus