User Question: I’m trying to make a parabolic arch, and I THINK my 3 point spline by points is parabolic, but I’m not sure. How can I verify this?
Buildz Von Doom: I have it on good authority that the 3 point spline by points is a true parabola, not an approximation.
User: How do I know you aren’t just a corporate stooge telling me what I want to hear?
BVD: Puny Worm, my word is Law! I have released my corporate flying monkeys to erase you! Before you are consumed by my simian maelstrom, take some small comfort in the parabolic excellence of the 3 point spline by independently verifying it using conic sections. Behold!
Ellipses, hyperbolas, and parabolas are geometrically created using sections of cones. Parabolas [B] are created by cutting a section parallel to the outer surface of the cone. Ellipses [A] are made from sections at a shallower angle than the outer surface, hyperbolas [C] from steeper angles. Often these curves look pretty similar, but they have different characteristics. Parabolas are the most common, often for structural applications.
To verify that your three point spline-by-points is in fact a parabola, start with any old cone and draw a reference line aligned with the surface like so:
Draw a void extrusion on the workplane of the reference line and drag it into the cone
Now you have a true parabolic edge on your cut cone. Draw a three point spline by points on it.
Now if you drag your ref line around (the whole line, so that you don’t change the angle) your spline will move with the cut, showing various sizes of parabolic curves.
You can zoom way in on these and see that the curve tracks right on top of the edge, not approximating the parabola.
For more on conic sections, check out Wolfram.
Enjoy the monkeys, Insolent Peasant!