Talk:Geometric continuity

This isn't Geometric Continuity... This article was listed at Wikipedia:1911 Encyclopedia topics/15, together with a link to the original 1911 article. That article is... horrid, but not about continuous functions, which this article is attempting to be about. Continuous functions are covered reasonably well at continuous function. The notion of continuity dealt with in the 1911 article, and apparently called "Geometrical Continuity" is:

• Somehow about conic sections, and passing from an ellipse, to a parabola, to a hyperbola, by moving through points at infinity, and some relation between that idea and imaginary numbers.

• Not an idea that's really talked about in modern mathematics, AFAIK.

I'm not sure what to do about this, but I'm making a note of what I saw... -GTBacchus 02:39, 30 October 2005 (UTC)Reply

Ok, here's a link to a modern definition of geometric continuity in the context of parametric curves. I guess I was wrong about the second bullet point above; although requiring that two segments of a parametric curve have proportional derivatives at the point where they meet is hardly what 1911 was talking about. -GTBacchus 02:47, 30 October 2005 (UTC)Reply

I've rewritten to reflect as much as I can of what the 1911 article is talking about. Anyone who wants to complain about the quality of Wikipedia can go read this 1911 article, which would not have passed modern muster. What a rambling piece of junk that was. If anyone can salvage more meaningful stuff out of the references there, please do. Alba 21:13, 1 February 2006 (UTC)Reply

I added this section after doing a computer graphics course, 'smoothness' may not be the best word to use here, and a picture would be nice if anyone's feeling arty. Davehodgson333 12:09, 26 December 2006 (UTC)Reply

G² continuity must not be defined in terms of curvature, it must be defined in terms of the second derivative. Curvature and second derivatives are not the same. In particular, the center of curvature at a point of inflection is indeterminate (i.e., either plus or minus infinity) though the second derivative there is zero.

G² continuity exists at a junction of two curves if those two curves meet with G¹ continuity and the second derivatives of each with respect to itself are equal at the junction. (Alternatively, but only for parametric curves, G² continuity exists if (1) two curves meet with G¹ continuity, (2) the direction of the second derivatives with respect to the parameter are equal at the junction, and (3) the proportion of the magnitudes of the two second derivatives with respect to the parameter at the junction equal the proportion of the magnitudes of the two first derivatives with respect to the parameter at the junction.)

David f knight (talk) 18:46, 7 September 2008 (UTC)Reply