Can anyone shed some light on an involute problem?


I was wondering if any of you kind fellows could help me with a problem...

I am currently writing some software for modelling involute gears in

3D: the 3D processing being handled by Microsoft's DirectX. My problems start when I come to cast light onto the model. To effectively light an object I need to know the normal vectors at each of the vertices. The parametric equation I am using to generate the tooth profile is:

Xi = r*(cos(t)+t*sin(t)) Yi = r*(sin(t)-t*cos(t))

Where t is the parametric variable and r is the radius of the gear's base circle.

Could anyone tell me how I could go about calculating the normal vectors? I think it has something to do with taking a unit vector to the tangent line (as calculated by differentiation?), but I'm clueless how to actually go about doing this. Any help would be much appreciated.

Many thanks,


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The tangent vector of the curve will be [dx/dt dy/dt] (assuming the format [x_component y_component]). The normal vector will be [dy/dt

-dx/dt]. So just take the derivative of Xi and Yi with respect to t analyticaly and use those values for t to get the normal vectors.

Hope this helps,


Reply to
Matthew Douglas Rogge

It'll take 3 vectors to get the normal vector; a tangent vector, a radial vector and a binormal vector.

So define a position or radial vector to a vertex with Xi & Yi. Then take the derivative to get a tangent vector. A binormal vector can then found by taking the cross product of the radial and tangent vectors. And a normal vector can be found from the cross product of tangent and binormal vectors. Normalize the normal vector to make a unit vector, i.e. divide each component by the vector magnitude.

Hope this helps.

Reply to
Jeff Finlayson

Perfect! Thanks guys: that was just what I was after. I should have asked on this group earlier, it would have saved me a fortune in strong coffee and chewed pencils.

All the best,


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