Ellipse Maths 2 - Tangents

Now we can find the point on an ellipse, given the angle around the ellipse we are positioned. What if we want to orientate our object to face in the direction it is moving as we increase or decrease the angle? Or what if we want our object to leave the ellipse and smoothly move onto another path, like a spline?

What we need is the tangent to the ellipse at a given angle. I knew this should be straight forward, but it took a while for my brain to slip back to A-Level maths mode and remember how to do this. Too much time writing Tools code I guess. Anyway, Calculus to the rescue!

So if P is the point on the ellipse that we defined in the previous post, we are looking for T the tangent on the line.

P = m_h * cos(t) * m_up + m_l * sin(t) * m_along + m_centre;

To find the tangent we want the derivative of this equation, that is the rate of change of P with respect to t ( the angle around the ellipse ). So we can simply differentiate w.r.t. t which is simple if you know that the differential of sin(t) is cos(t) and the differential of cos(t) is -sin(t). Giving us

T = - m_h * sin(t) * m_up + m_l * cos(t) * m_along;

Note, that we've lost m_centre, because that is a constant.
And that is it, here is the full method

Vector3 TangentAt(float t) const
{
float c = cos(t);
float s = sin(t);

return - m_h * s * m_up + m_l * c * m_along;
}