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In practice the intersecting plane must be tangent to the inner equator of the torus. It is generated as a Cassini’s oval with the further requirement that. īernoulli’s lemniscate is, in turn, a special case of a Cassini’s oval.
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If the focal distance is, the Cassini’s ovals parameters, with reference to the torus intersected, have the values and. This definition recalls that of the ellipse but here we must use the product of the distances instead of the sum. The name comes from the ancient Greek word “ σπειρα” for torus.įall in this category Cassini’s ovals and Bernoulli’s Lemniscates.Ĭassini’s ovals are spiric sections in which the distance of the cutting plane to the torus axis equals the radius r of the generating circle.Īn interesting alternative definition of a Cassini’s oval is that of the set of points P such that the product of their distances to two fixed points F 1 and F 2 is constant: The toric section generated by an intersecting plane that is parallel to the torus axis (or perpendicular to the torus equatorial plane) is called a spiric section. It must be noted that, if some central conic section generates circles, not all central toric sections are circles as in the following examples. 03b – Position of the intersecting plane to generate the Villarceau circles Central toric sections – other sections The last (less banal) case produces two circles that are called Villarceau’s circles.įig. the intersecting plane touches the torus in two isolated points.the intersecting plane is perpendicular to the equatorial plane.the intersecting plane is the equatorial plane.The central toric sections and the Villarceau’s circlesĪ toric section in which the cutting plane passes through the center of the torus is called central toric section.Ĭentral toric sections can be circles in the following cases: Amongst them Villarceau’s circles, Cassini’s ovals, Bernoulli’s lemniscates and the Hippopedes of Proclus. In this section we’ll shortly describe and present some particular toric sections that also have an historical relevance. Among them, one can distinguish Cassini’s curves (in particular, Bernoulli’s lemniscate). The corresponding curves are usually called “the spiric sections of Perseus” after their discoverer (circa 150 BC). The variety of the toric sections is quite rich and the study of the case when sectioning planes are parallel to the symmetry axis of the torus dates back to antiquity. In the following lines (if not otherwise specified) we’ll always assume that (to avoid self-intersections), and that, in Cartesian coordinates, the equatorial plane is the xy plane and the axis of rotation is the z axis.Ī toric section is the analogue of a conic section as it is the intersection curve of a torus with a plane just as a conic section is the intersection curve between a conical surface and a plane.īut whilst conic sections have a (deserved) renown also due to their multiple connections with many fundamental problems of (classical) physics, toric sections are often relegated to the realm of mathematical curiosities and exotic curves. The two radii r and R are so the parameters that identify the torus’ shape.
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The torus surface is generated by rotating a circle with radius r around an axis coplanar with the same circle, following a second circle of radius R. Some notion of goniometry and of tridimensional analytic geometry. In the article only elementary algebra is used, and the requirements to follow it are
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In this article I’ll discuss some properties of this curve, investigate its differences with the most renowned conic section, show how to build its general quartic equation, explore how a toric section can also be generated by intersecting a cylinder with a cone and finally describe how it is possible to represent it in the 3D Graphics view of Geogebra. Even if both surfaces are rather simple to define and are described by rather simple equations, the toric section has a rather complicated equation and can assume rather interesting shapes. The curve of intersection of a torus with a plane is called toric section. 01 – The torus-Plane Intersection simulation with Geogebra Overview