Drawing Curves with Conic Sections: A Guide

[Bruno Tolentino]

I'd like of draw any curve using combination of line circle, elipse, parabola, hyperbola and straight. Of course several curves can't be designed with 100% of precision using just conic curves, but, can to be approximated.

Acttualy, I don't want to reproduce a curve already designed but yes produce a curve from of white sheet.

I think that the most intuitive idea is choice two or more points (let's choose just two points, the start and the end) and in these points, specify the line tangent, see:

So, how can I complete the path from A to B with a continuous, smooth and conic curve?

 

[Nidum]

You don't say whether you want a mathematical solution or just a bit of practical drawing .

If the latter get yourself some simple CAD software . Most versions have a comprehensive range of curve fitting functions . Not only simple curve fitting but often several varieties of spline fitting as well .

 

[SteamKing]

I'd like of draw any curve using combination of line circle, elipse, parabola, hyperbola and straight. Of course several curves can't be designed with 100% of precision using just conic curves, but, can to be approximated.

Acttualy, I don't want to reproduce a curve already designed but yes produce a curve from of white sheet.

I think that the most intuitive idea is choice two or more points (let's choose just two points, the start and the end) and in these points, specify the line tangent, see:

So, how can I complete the path from A to B with a continuous, smooth and conic curve?

With the two arbitrary points as shown and the tangents at each, you can construct a circular arc connecting A and B.

These types of constructions are essentially geometric in principle and used to be taught when drafting was a manual skill, not dependent on the use of CAD.

For the problem as shown, you want to draw two additional lines, one perpendicular to each tangent at points A and B. The intersection of these two perpendiculars will lie at another point C, which will be the center of the circular arc connecting A and B.

 

[ChrisVer]

There is an infinite set of curves you can draw that connect the two points A,B
And at each point inbetween the curve can be given the value of tangent on that curve and the curvature of the curve on that point..