If you project a helix onto a 2D plane from a perspective normal to its axis, what type of curve is the result, and what formula describes it?

 

Another way of putting it: If I want to draw a side-view of a twist drill bit, how do I draw the flute?

 

Thanks!

It should just be a sinusoid. If you imagine a plane with the helix axis normal to it being moved along the helix, the intersection of the helix and the plane will move around in a circle at a constant rate.

 

The distance of the intersection from the axis of the helix is constant, but the displacement expressed as as two orthogonla vector components, i.e. projections onto two orthogonal planes will be two orthogonal sinusoids.

 

You can pick any plane and the projection will be a sinusoid with the phase depending on the plane chosen.

If you project a helix onto a 2D plane from a perspective normal to its axis, what type of curve is the result, and what formula describes it?

 

Another way of putting it: If I want to draw a side-view of a twist drill bit, how do I draw the flute?

 

Thanks!

First of all, state the problem accurately! The projection of a helix onto a plane normal to the axis is a circle. Apparently you want a helix projected onto a plane parallel to the axis. That projection will be a cycloid.

 

http://mathworld.wolfram.com/Cycloid.html

I assumed he meant parallel, but as for the cycloid... Why?

 

I'm not sure here, but from observing my trusty spring, it looks as if thats just the general case for the projection onto any plane, and just as the circle is one extreme, for this other extreme, we can specifically say its sinusoidal.

HallsofIvy: I didn't say the plane was normal, I said the perspective was normal. But I'll allow as how it was ambiguously stated, though not inaccurate. The 'other way of putting it' should have removed the ambiguity. And I see no way the continuous smooth curve could possibly be a cycloid. The first reply makes sense to me; yours does not.

 

Now, successive question: can a sine wave, or portion thereof, be closely approximated using cubic or quadratic Bézier curves or elliptical arcs? I want to represent the curve in SVG and those are the only curve commands available. So far all the SVG representations of sine curves I've found have been done in terms of lots of short line segments.