Orthoprojection of circle onto a plane

[geoffrey159]

Homework Statement

Show that the orthoprojection of a circle onto a plane is an ellipse

Homework Equations

Let's say the circle ${\cal C}$ of center $O$ and radius $R$ lies on plane $P$ and we want to orthoproject ${\cal C}$ onto $P'$

The Attempt at a Solution

We can say that up to a translation of vector $\vec{O\pi_{P'} (O)}$ , we can orthoproject ${\cal C}$ onto plane $P''$ such that $P$ and $P''$ intersect over line $D$, and its center $O \in D$.

Assume that $D$ is directed by unit vector $\vec u$. We can find an orthonormal basis of $\vec P$, say $(\vec u, \vec v)$, such that $(\vec u, \vec w = \text{rot}_{\vec u, \theta} (\vec v))$ is an orthonormal basis of $\vec {P''}$, where $\theta = \angle (P,P'')$.

Then $M(x,y) \in P \Rightarrow \pi_{P''}(M) = (x \vec u,\vec u) \vec u + (y\vec v,\vec {w}) \vec w = x \vec u + y \cos (\theta) \vec w$

And then $M \in {\cal C} \iff x^2 + y^2 = R^2 \iff \frac{x^2}{R^2} + \frac{(y\cos(\theta))^2}{R^2 \cos(\theta)^2} = 1 \iff \pi_{P''}(M)$ is located on an ellipse of $P''$

Is this working for you ?

 

[mfb]

Looks right.
I would explicitely replace y cos(θ) by w for the last step.

 

[geoffrey159]

Ok, thanks !