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It seems that an 'arc' is a line-segment mapped onto the surface of a sphere (although I don't know if that name still holds if the segment wraps around the sphere more than once, i.e., if the angle subtended by the arc is $>2\pi$). Is there a name for higher-order polynomials mapped onto the surface of a sphere?


Could this be related to geodesics? Does there exist a generalization of geodesics parallel to the polynomial generalization of lines? I'd also appreciate some pointers on where to look for more information on the idea of doing flat math on the surface of a sphere: e.g., if you choose a point on the surface of a sphere and define some arbitrary orthonormal basis set to create 2d a coordinate system on the surface, then you can follow some set of rules for how that coordinate system changes as you plot out a function. Sure the coordinate system will collide with itself, but that's fine as long as any point of a function knows what its own local coordinate frame is. Does that description even make sense?

Edit to clarify mapping:

In Euclidean space, you can define a quadratic spline segment with two linear segments that share an endpoint. If we parametrize the spline with $t$ and have points $P(0)$, $K(.5)$, and $P(1)$, then $P(t) = (1-t)^2P(0) + 2t(1-t)K(.5) + t^2P(1)$ describes a quadratic curve that interpolates $P(0)$ and $P(1)$ and has first derivatives defined by $K(.5)$.

I can define an arc on the unit sphere with two orthonormal vectors and an angle (so that I can define arcs subtending $\pi$ radians). If those vectors are $A(0)$ and $B(0)$ with angle $\theta$, then my "first order spherical spline is defined as $Q(t) = \cos(t\theta)A(0) + \sin(t\theta)B(0)$. I can use the same sort of approach as used to make a quadratic spline from two linear splines by setting $A(1) = Q(1)$ with $B(1)$ and $\theta_2$ as free parameters to define the curve (although $B(1)$ must be orthogonal to $A(1)$). There's an extra degree of freedom in the spherical case that must be dealt with (how many times the $A(0),B(0)$ plane flips around as it interpolates onto the $A(1),B(1)$ plane), but the mapping is pretty straightforward. So it would seem that this is a quadratic curve on a spherical surface. It looks cool. Does it have a name?

I can find references to Bernstein-Bezier polynomials on a sphere, but this seems to be something different.

Animated curve example

In this example, $A(0)$ is red, $B(0)$ (usually covered) is pink, $A(1)$ is green, $B(1)$ is cyan, the final endpoint is blue. Yellow and orange represent the plane through which $A(0),B(0)$ rotate to interpolate onto $A(1),B(1)$. I change the curve parameters while it rotates.

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What sort of mapping on the sphere did you have in mind? – J. M. Apr 19 '11 at 15:32
@J.M. I edited to clarify. – JCooper Apr 19 '11 at 15:54
+1 for nice graphic! – Grumpy Parsnip Apr 19 '11 at 18:04
Although it might not be exactly what I'm doing, if you grab a line segment by its endpoints and 'drape' it on a sphere such that while you're 'draping' you place the line directly down against the surface normal of the sphere, you get an arc going around the circumference of the sphere. If you were to lift a segment of parabola off of a plane and do the same, what would you have? – JCooper Apr 19 '11 at 19:58
up vote 4 down vote accepted

One way of approaching this question leads you into a very cool area of math called algebraic geometry. Unfortunately, I know next to nothing about algebraic geometry. I'll try to give you a few ideas to get you started, but it would be great if people who know more about the subject could jump in with better answers!

Before we try to figure out what "a parabola on a sphere" is, let's make sure we understand what a parabola on a plane is.

The general equation for a parabola is $$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0,$$ with the restriction that $B^2 - 4AC$ must be zero. If you set $B$ and $C$ to zero, and $E$ to negative one, you'll see that the more familiar equation $$Ax^2 + Dx + F = y$$ is just a special case of the general one.

If you drop the restriction that $B^2 - 4AC$ must be zero, the equation $$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$ can describe a parabola, an ellipse, or a hyperbola. These shapes are called "quadratic curves," because they're described by quadratic equations---polynomial equations where the highest power is two.

Instead of just trying to generalize parabolas to a sphere, let's try to generalize all the quadratic curves to a sphere.

A sphere, like a plane, is a two-dimensional surface, but it's easiest to think of a sphere as floating in three-dimensional space. For example, we can think of a sphere as the set of solutions to the equation $$x^2 + y^2 + z^2 = 1.$$ In order to put the sphere and the plane on equal footing, let's also imagine our plane floating in three-dimensional space. For example, we can think of the plane as the set of solutions to the equation $$z = 0.$$

In three dimensions, a quadratic curve on the plane is described by a system of two equations: $$\begin{cases} Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \\ z = 0. \end{cases}$$ The first equation is the one we saw before, and the second equation says that the curve has to lie on the plane.

In two dimensions, the first equation was the most general possible quadratic equation. In three dimensions, we can make the first equation more general by adding all the possible $z$ terms: $$\begin{cases} Ax^2 + Bxy + Cxz + Dy^2 + Eyz + Fz^2 + Gx + Hy + Iz + J = 0 \\ z = 0. \end{cases}$$ This doesn't change the shapes we're describing at all, because substituting the second equation into the first equation makes the $z$ terms disappear.

Now, if someone asks us what a quadratic curve on a plane is, we have a really nice answer.

In three dimensions, a quadratic curve on a plane is described by a system of two equations: a quadratic equation, and the equation for a plane.

And if someone asks us what a quadratic curve on a sphere is, we won't have to think very hard to come up with an answer for that too!

In three dimensions, a quadratic curve on a sphere is described by a system of two equations: a quadratic equation, and the equation for a sphere.

In fact, we might as well go right ahead and say that...

In three dimensions, a quadratic curve on SURFACE is described by a system of two equations: a quadratic equation, and the equation for SURFACE.

In two dimensions, the shape described by a single quadratic equation is one-dimensional, so it's called a quadratic curve. In three dimensions, the shape described by a single quadratic equation is two-dimensional, and it's called a "quadratic surface." Using this terminology, we can make our definition of a quadratic curve on a surface even prettier.

In three dimensions, a quadratic curve on SURFACE is the intersection between SURFACE and a quadratic surface.

Now that we've decided what quadratic curves on a sphere are, the obvious thing to do is to start drawing them! Personally, I find quadratic curves a bit dull, but cubic curves look really cool. (Wait, what's a cubic curve? Figure it out!)

If you get tired of drawing, here are some puzzles to chew on.

  1. The quadratic surfaces $x^2 + 3y^2 - z^2 - \tfrac{1}{2} = 0$ and $2y^2 - 2z^2 + \tfrac{1}{2} = 0$ cut out the same curve when you intersect them with the sphere $x^2 + y^2 + z^2 = 1$. Why? Find all the quadratic surfaces that cut out this curve on the sphere.

  2. The system $$\begin{cases} x^3y + xy^3 + xyz^2 - xy - yz + x = 0 \\ x^2 + y^2 + z^2 = 1 \end{cases}$$ describes a quadratic curve on the sphere, even though the first equation isn't quadratic. Why? Find all the ways you can change the first equation and still get the same curve on the sphere.

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Cool. I'm glad to see an answer after so long. The field of Algebraic Geometry is definitely new to me. It doesn't actually give a name to the generalization of geodesics to higher-degree polynomials, but maybe it doesn't need a name. I can't easily render curves described this way at the moment. Do you happen to have any intuition as to how these might compare with defining a portion of a curve as a parametrized interpolation between two arc segments? – JCooper Dec 31 '11 at 6:09
The program surfex can render algebraic curves. The output is really nice, but installing it is a bit of a nightmare, and using it isn't totally straightforward. Let me know if you'd like any help with it! – Vectornaut Jan 2 '12 at 1:58
Judging from your picture, parameterized interpolation between arc segments doesn't give quadratic curves on the sphere, because a general quadratic surface is an ellipsoid, a paraboloid, or a hyperboloid, and the curves you drew don't look like the intersection of the sphere with one of those. Your interpolated curves might be higher-order polynomial curves, but my gut feeling is that they aren't. Then again, my gut feelings in this area probably aren't worth much! – Vectornaut Jan 2 '12 at 2:08
@JCooper: Since you mentioned geodesics again, let me bring up something I didn't talk about in my answer. In geometry, there are two main ways to generalize the concept of a "straight line": the algebra way and the calculus way. The algebra way starts with the observation that a line is described by a linear equation, and leads into algebraic geometry. The calculus way starts with the observation that a line (roughly speaking) has second derivative zero, and leads into an area called Riemannian geometry. CONTINUED... – Vectornaut Jan 2 '12 at 2:25
...CONTINUED The idea of a "geodesic" is part of Riemannian geometry. I'm pretty sure there is a "second-order" generalization of geodesics, but it might require a little more background to understand, and I'm not sure what the best way of explaining it is. Tell me if you'd like to know more! – Vectornaut Jan 2 '12 at 2:30

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