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The quarter of Viviani's curve that lies in the all-positive quadrant of 3D space cannot be represented exactly by a regular Bézier curve of any degree.. However, it can be represented exactly by a 3D rational Bézier segment of degree 4, and there is an infinite family of rational Bézier control points generating that segment.
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Viviani's curve as intersection of a sphere and a cylinder. In the case = +, the cylinder and sphere are tangential to each other at point (,,). The intersection resembles a figure eight: it is a closed curve which intersects itself. The above parametrization becomes
The three surfaces intersect at the point P with those coordinates (shown as a black sphere); the Cartesian coordinates of P are roughly (1.0, −1.732, 1.0). Cylindrical coordinate surfaces. The three orthogonal components, ρ (green), φ (red), and z (blue), each increasing at a constant rate. The point is at the intersection between the ...
The sum of the angles of a spherical triangle is not equal to 180°. A sphere is a curved surface, but locally the laws of the flat (planar) Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is only slightly more than 180 degrees. A sphere with a spherical triangle on it.
In the third case, the sphere and cylinder touch each other at one singular point. The intersection curve is self-intersecting. If the cylinder and sphere have the same radius and the midpoint of the sphere is located on the axis of the cylinder, then the intersection curve consists of singular points (a circle) only.
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