The normal at (x1, y1) is $a^2$*x/x1 - $b^2$y/y1 = $a^2$ - $b^2$. In parametric form: ax/cos(theta) - by/sin(theta) = $a^2$ - $b^2$. In slope form: y = mx - m$\frac{a^2-b^2}{sqrt}$($a^2$ + $b^2$$m^2$). From an external point, at most 4 normals can be drawn (contrast: 3 for parabola). The normal at P bisects the angle between the focal radii SP and S'P (the internal angle), while the tangent bisects the external angle. This is the reflection property: a ray from one focus reflects through the other focus. If the normal at eccentric angle theta meets the major axis at G, then CG = $e^2$acos(theta). The evolute (envelope of normals) is the astroid (ax)^$\frac{2}{3}$ + (by)^$\frac{2}{3}$ = (a^{2-b}^2)^$\frac{2}{3}$.