# Support function of an ellipse

I defined the support function $h_A:R^n→R$ of a non-empty closed convex set $A\subseteq \mathbb{R}^n$ as $$h_A(x)= \sup\left\{x⋅a |\ a \in A\right\}$$

Everything I know about this topic I found it. I have to calculate the support function of an ellipse $$\text{E=\left\{(x,y) \in \mathbb{R}^2 \quad|\quad \frac{x^2}{a^2}+\frac{y^2}{b^2}\le 1 \right\}}$$

For $x=a\cos\theta ,\ y=b\sin\theta, \ 0\le\theta\lt2\pi\$, how can calculate $h_E(\theta)$?

The result should be $h_E(\theta)=\left(a^2\cos^2\theta+b^2\sin^2\theta \right)^\frac12$ but my calculations lead me to have $h_E(\theta)=\sup\{(a\cos\theta,b\sin\theta)⋅(\cos\theta,\sin\theta)\}=a\cos^2\theta+b\sin^2\theta$.

Where is the error?

• Please change the variable over which the supremum is taken so that it's different from the function's variable. At present the definition of $h_A$ isn't clear. – Micapps Jun 5 '16 at 10:22
• I've done. Thank you for remark. Is clearer now? – J.Doe Jun 5 '16 at 10:33

The main problem you have is giving the same name to different things: $\theta$ means two different things, and so does $a$.
The parametric equation $x=a\cos t$, $y=b\sin t$, leads to $$h_E(\theta)=\sup_t (a\cos t,b\sin t)\cdot (\cos\theta, \sin\theta)$$ Observe that the dot product can just as well be written as $$(\cos t,\sin t)\cdot (a\cos\theta, b\sin\theta)$$ which is simply the projection of $(a\cos\theta, b\sin\theta)$ onto the direction determined by $t$. The maximal possible value of scalar projection is the length of the vector, hence $$h_E(\theta)=|(a\cos\theta, b\sin\theta)| = \sqrt{a^2\cos^2\theta+b^2\sin^2\theta}$$
A general hyper-ellipoid is the affine image of the unit-ball. This can be written as $$\mathcal E := \{Ax + c \text{ s.t } x \in \mathbb B_n\}$$, where $$\mathbb B_n := \{x \in \mathbb R^n \text{ s.t } \|x\|_2 \le 1\}$$ is the unit-ball in $$\mathbb R^n$$, $$A : \mathbb R^n \rightarrow \mathbb R^n$$ is a linear transformation and $$c \in \mathbb R^n$$, is the center of the ellipsoid. Now, one computes the support function $$\sigma_{\mathcal E}$$ of $$\mathcal E$$ as
$$\begin{equation} \begin{split} \sigma_{\mathcal E}(z) &= \sup\{\langle z, y\rangle \text{ s.t } y \in \mathcal E\} = \sup\{\langle z, Ax + c\rangle \text{ s.t } x \in \mathbb B_n\} \\ &= \langle z, c\rangle + \sup\{\langle A^Tz, x\rangle \text{ s.t } x \in \mathbb B_n\} = \langle z, c\rangle + \sigma_{\mathbb B_n}(A^Tz) = \langle z, c\rangle + \|A^Tz\|_{(2^*)}\\ &= \langle z, c\rangle + \|A^Tz\|_2. \end{split} \end{equation}$$
In your particular case $$c = 0 \in \mathbb R^n$$, and $$A$$ is an orthonormal matrix with entries $$\pm \sin(\theta), \pm \cos(\theta)$$. Figure out the the precise values for these entries, plug them into the formula i derived above, and you're done.