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equation for conic curve

equation for conic curve

equation for conic curve

(OP)
Any cad system appropriate for aerospace has a conic curve function, where you can create a conic curve and change its shape with a rho parameter.  Rho = .5 gives a parabola, and greater or lesser gives hyperbolas and ellipses, respectively.
What I am looking for is a cartesian equation incorporating the rho value that will give me point coordinates.
This will help me to create a spreadsheet to determine the volume of a propellant tank with conic section domes.
Any leads on such an equation?

RE: equation for conic curve

do you mean like the equation of an ellipse ...
(x/a)^2+(y/b)^2 = 1

RE: equation for conic curve

not knowing how 'rho' is used, one could only guess. By looking this up on the Internet, I found the general form for a conic:
Ax*x + Bxy + Cy*y + Dx + Ey + F = 0

Which type of curve you get depends on the value of the
discriminant D, in which D is D==B*B-4*A*C

 If D=0 then curve is parabola, 2 parallel lines, 1 line or no curve.
 If D<0 then curve is ellipse, circle, point or no curve.
 If D>0 then curve is hyperbola or 2 intersecting lines.

 You could artificially define another, auxiliary
 constant called G, for instance, in which G==D+0.5.
 In that way, when G=0.5, D=0, means you get parabola, 2
 parallel lines, 1 line, or no curve.

In any case, you can see by doing this that you really do not necessarily get 1 of 3 curve types, parabola, ellipse, or hyperbola just by varying G or D--you could get one of the other object types: 2 parallel lines, 1 line, no curve, point, or 2 intersecting lines. There is quite a bit of ambiguity (and flexibility) built into the General Conic Equation above. Perhaps the CAD system you are using defines its own conic equation--shouldn't the Help manuals have this information?

RE: equation for conic curve

According to the MMachinery's Handbook volume of an ellipsoid is (4/3)Pi(a)(b)(c), where a,b,and c are the radii of the three axes of revolution.

RE: equation for conic curve

(OP)
Thanks for the help.  I have gone through approaches similar to your suggestions, but the problem is holding tangencies at the ends of the curve.  I think the answer is to define a pseudo conic curve with a Bezier curve, if I ever learn how to do it.

RE: equation for conic curve

That was one of the points of the ellipsoid equation. If you cut the ellipsoid in half the tangents of the surface will blend into a cylinder with no discontinuity of slope. Using it in any other way would make for some more difficult math. But that is exactly what you want. I belive for your pupose b=c.

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