Numerically finding the projected area of a bullet

Question

Suppose I have a bullet as shown below where the measurements are in units of bullet diameters (this thing is 3 dimensional, so imagine rotating it about the x axis here)

If this bullet were to be tilted upwards by an angle θ, how could I numerically find its projected area?

I'm trying to find the area that such a bullet would present to the air as it moves through it and so if it is not tilted away from the direction of motion this area is simply a circle. I know for small tilts, it will simply present the projected area of a cylinder but I am unsure about how to deal with tilts large enough that one needs to care about the tip of the bullet for purposes of finding the area. Anyone have ideas about how to deal with this?

Is the plane that you are projecting into always perpendicular to the bullet in it's above position? — Alexandru Barbarosie, Oct 29 '17 at 01:37
No, in general I would like the plane to make any angle with respect to the bullet. — Mason, Oct 29 '17 at 07:19
Yeah, I get that the plane will in different position with respect to the bullet (depending on the angle), but what will be that plane? — Alexandru Barbarosie, Oct 29 '17 at 13:23
Simple approach is to discretize surface, project vertices, and take convex hull of projected vertices. — Ante, Oct 29 '17 at 16:12

score 0 · Answer 1 · 2017-10-29T18:35:51.200

Hint:

The boundary curves of the bullet are the apparent outline of the inner surface of a self-intersecting torus. They can be found by expressing that the normal vector is parallel to the projection plane.

With z being the axis of the bullet, the parametric equation of the surface is

x= (R + r sinφ) cosΘ
y= (R + r sinφ) sinΘ
z=      r cosφ

and the normal is obtained by setting R=0,

x= r sinφ cosΘ
y= r sinφ sinΘ
z= r cosφ

Now for some projection plane with a normal in direction (cosα, 0, sinα), the outline is such that

r sinφ cosΘ cosα + r cosφ sinα = 0.

From this you can draw Θ as a function of φ or conversely and construct points along the curve.

When α increases, the tip of the bullet starts entering the ellipse resulting from the projection of the basis of the cylindre. This ellipse corresponds to the angle φ such that z=0.

The surface is known as a lemon shape: http://mathworld.wolfram.com/Lemon.html

Numerically finding the projected area of a bullet

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