The sphere (x-1)^2+(y-1)^2+(z-1)^2=1 is lit by rays parallel…

The sphere $(x-1)^2+(y-1)^2+(z-1)^2=1$ is lit by rays parallel to $x=y=z$. Find the shadow on $xOy$.

Step 1: Tangent rays

The ray through $(u,v,0)$ is $(u+t,\,v+t,\,t)$. Its distance from $C(1,1,1)$ is found from

\[ \{u-1,v-1,-1\}\times\{1,1,1\}=\{v,-u,u-v\},\qquad d^2=\frac{u^2+v^2+(u-v)^2}{3}. \]

The boundary of the shadow consists of the grazing rays, where $d=1$: $2x^2-2xy+2y^2=3,\quad z=0$.

The shadow itself is the region $2x^2-2xy+2y^2\le3$.

Step 2: The ellipse

Rotate by $45^\circ$: $x=\tfrac{u-v}{\sqrt2}$, $y=\tfrac{u+v}{\sqrt2}$. This gives $u^2+3v^2=3$, that is

\[ \frac{u^2}{3}+v^2=1 . \]

This is an ellipse centered at $O$. Its semi-axes are $\sqrt3$ along the line $y=x$ and $1$ along $y=-x$. The center $O$ is the shadow of $C$, because $C-\{1,1,1\}=O$. The minor semi-axis equals the radius. The major one is stretched by $\tfrac1{\cos\theta}=\sqrt3$, where $\theta$ is the angle between the rays and $Oz$.