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A point source emits light at a circular disc (thickness negligible), and a shadow is left on a wall (XY plane) behind and parallel to the disc. The Z component of distance between the point source and the disc is 'a' units. The Z component of distance between disc and wall is 'b' units. The line joining the point source and the centre of the disc lies on the YZ plane, intersects the wall (XY plane) at origin, and intersects the Z axis at angle theta. If the radius of the disc is r, find the equation of the perimeter of the shadow projected on the XY plane enter image description here

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I think you can use similar triangles to show that the image is a circle centred at the origin and calculate the radius. Take a point $P$ on the circle. Call the position of the light source $S$, the centre of the circle $C$, the origin $O$ and let the image of $P$ in the $XY$ plane be $Q$.

Then $CP$ is parallel to $OQ$ because the circle lies in a plane parallel to the $XY$ plane, so $\triangle SOQ$ is similar to $\triangle SCP$. The similarity ratio is $\frac {SO}{SC}$ and the distance $OQ = r\frac {SO}{SC}$.

Mark Bennet
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  • Your cone is not a right circular cone - your circle is not perpendicular to the line $SCO$. The fact that it lies in a plane parallel to the $XY$ plane means that the plane defined by the lines $SCO$ and $SPQ$ cuts the parallel planes in parallel lines. This enables $OQ$ to be expressed in a form independent of the choice of $P$, and this proves the assertion that the image is a circle with centre $O$. – Mark Bennet Jun 01 '13 at 11:45
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    Then you will find that the intersection of $SC$ with the $XY-\text{plane}$ will be the centre of the circle. I didn't use the fact that $O$ was the origin, just that $SCO$ was a straight line. So replace $O$ with the point which does make a straight line - the computations may be a little messier, though. – Mark Bennet Jun 01 '13 at 12:09
  • It's still a circle if the original circle lies in a plane parallel to the plane containing the shadow. – Mark Bennet Jun 01 '13 at 12:33