Tag: Triply Orthogonal Surfaces

Quadrics

I have written about triply orthogonal surfaces twice before here, in the case of spheres and of cyclides, thus omitting the best known examples, namely that of quadrics. A quadric is for space what a conic is for the plane, and, to warm up, here are some conics ⎯ ellipses and hyperbolas ⎯⎯, all with the same focal points.

Confocal 01

That they all meet orthogonally is not difficult to see, one can either use the geometric definition of these conics as curves whose points have constant distance sum/difference to their focal points, or an algebraic description as level sets of quadratic polynomials.

In the plane, there is one other kind of conic, namely the parabola, and here a single family of confocal parabolas provides us already with a doubly orthogonal system of curves:

Parabolas 01

While the images are pretty, there is nothing astonishing happening here: Any reasonable curve family will allow you to find orthogonal trajectories, and the pigeonhole principle or one’s belief in the pre-established harmony of the universe will force cases where both curve families are simple.

Not so in dimension 3: A surface family in space only belongs to a triply orthogonal system of three surface families if it satisfies a rather complicated partial differential equation, which I believe was first found & used by Jean Gaston Darboux.

But again there are simple cases, and the algebraic argument that establishes the orthogonal hyperbolas and ellipses above also establishes that their 3-dimensional analogues form a triply orthogonal system of surfaces.

Quadrics

Here you can see all three general kinds of quadric surfaces: An ellipsoid, and the two different hyperboloids. The green one is the so-called single hyperboloid: it continues through the ellipsoid and has only one component. The yellow one is the double hyperboloid and has two components. I have mentioned the single hyperboloid before in connection with Brianchon’s theorem.

One reward for all these efforts to have them meet orthogonally is that one can see immediately the curvature lines of them, because a theorem of Pierre Charles François Dupin (not to be confused with Edgar Allan Poe’s detective C. Auguste Dupin) says that in triply orthogonal systems, two of the surfaces always meet in a curvature line of the third surface. The following image illustrates this for the ellipsoid: I have clipped the hyperboloids using a slightly larger (invisible) ellipsoid. This looks like it is complicated to make, but in fact requires only a few lines of code in PoVRay, a text based ray tracer that allows you to do constructive solid geometry and simple math, besides many other things.

Curvaturelines

Hermetic Geometry (Spheres II)

Objects in space can be either separate, intersecting, or touching. Spheres always intersect in circles, and the most fundamental case is when the angle of intersection is 90º.

Colorcircles

Let’s look at intersecting circles first. Somewhat surprisingly, two circles that meet at one point at a right angle also meet at the other intersection point at a right angle. This observation allows to construct two families of circles such that any two circles from both families meet at right angles. The first family consists of all circles that touch the x-axis at the origin, and the second family consist of all circles that touch the y-axis at the origin.

For what’s to come, let’s also look at these circles just in the first quadrant.

Halfcircles

The same procedure works in space with now three families of spheres. Each family consist of all those spheres that touch one of the coordinate planes at the origin. Viewing these in the first octant only and head-on, i.e. in the direction of the main space diagonal, shows patterns that disguise their origin quite well.

Triply1

Again, like in Spheres I, the alienation becomes extreme when we pretend that the objects are part of some large piece of architecture. The walls of the facade show the same pattern as the intersecting circles.

Triply2small

This way geometry becomes hermetic. Instead of explaining a theorem, the statement is disguised in pure appearance.