Are they perfect ellipses?

So while golfing, I noticed that the shadows of my golf balls looked like perfect ellipses. However, I thought about it, and this seems like a very sketch claim. If we consider the ground to be a perfectly flat plane, and the sun to be a point source infinitely far away, is the shadow created perfectly elliptical? Or is it slightly skewed away from the ball (personally this is my thought)? Or is this skew virtually 0.

Note by Trevor Arashiro
5 years, 5 months ago

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This is a perfect ellipse as ellipse is formed by stretching a circle.

- 5 years, 5 months ago

I would tend to agree. It is equivalent to the projection of a circle onto a plane, which is indeed an ellipse. Since the darkness and hence edge "sharpness" of the shadow varies with its distance from the ball there might appear to be some skewing of the ellipse, but I think that's just an illusion.

Just don't get too distracted by the ball's shadow, @Trevor Arashiro ; you'll end up taking a huge divot. :P

- 5 years, 5 months ago

Well, I had some more time to think about this, and I had assured myself that it wasn't a perfect ellipse. Then I came here and saw this.

I imagined that if the light source was infinitely far away but was just as far from the plane as the tip of the ball, the shadow's tip would be infinitely far from the ball. However, if we looked at where the so called "ellipse" was widest, it would be a finite distance $x$ away from the ball. If we take a slightly less extreme view of this, it would still be skewed right.

- 5 years, 5 months ago

Since the sun is in essence an infinite light source we can consider the incoming rays to all be parallel. If they were parallel to the plane of the ground then the shadow of the ball would only be seen on a vertical surface place behind the ball. If the incoming rays are at an angle then the relative proportions of the ball, (which as an obstacle to the light is the same as a disc), will be projected onto the ground linearly to create a perfect ellipse. I could still be wrong, so keep looking at shadows to find an exception. I'm surprised that an extensive google search has come up empty; it seems like such a natural question to contemplate.

- 5 years, 5 months ago

Come to think of it, the one case that I was thinking of was the extraneous case. We're simply looking at the cross section of a plane passing through a cone at an angle which is of course an ellipse.

- 5 years, 5 months ago

Nice observation! Good meat to ponder upon :P

- 5 years, 5 months ago