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2. Solid of revolution - Wikipedia

   en.wikipedia.org/wiki/Solid_of_revolution

   Two common methods for finding the volume of a solid of revolution are the disc method and the shell method of integration.To apply these methods, it is easiest to draw the graph in question; identify the area that is to be revolved about the axis of revolution; determine the volume of either a disc-shaped slice of the solid, with thickness δx, or a cylindrical shell of width δx; and then ...

3. Torus - Wikipedia

   en.wikipedia.org/wiki/Torus

   Poloidal direction (red arrow) and toroidal direction (blue arrow) A torus of revolution in 3-space can be parametrized as: [2] (,) = (+ ⁡) ⁡ (,) = (+ ⁡) ⁡ (,) = ⁡. using angular coordinates , [,), representing rotation around the tube and rotation around the torus' axis of revolution, respectively, where the major radius is the distance from the center of the tube to the center of ...

4. Point groups in four dimensions - Wikipedia

   en.wikipedia.org/wiki/Point_groups_in_four...

   A hierarchy of 4D point groups and some subgroups. Vertical positioning is grouped by order. Blue, green, and pink colors show reflectional, hybrid, and rotational groups.

5. Toroid - Wikipedia

   en.wikipedia.org/wiki/Toroid

   A toroid using a square. A torus is a type of toroid.. In mathematics, a toroid is a surface of revolution with a hole in the middle. The axis of revolution passes through the hole and so does not intersect the surface. [1]

6. Surface of revolution - Wikipedia

   en.wikipedia.org/wiki/Surface_of_revolution

   A portion of the curve x = 2 + cos(z) rotated around the z-axis A torus as a square revolved around an axis parallel to one of its diagonals.. A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) one full revolution around an axis of rotation (normally not intersecting the generatrix, except at its endpoints). [1]

7. Rotations in 4-dimensional Euclidean space - Wikipedia

   en.wikipedia.org/wiki/Rotations_in_4-dimensional...

   The conjugate of that eigenvalue is also unity, yielding a pair of eigenvectors which define a fixed plane, and so the rotation is simple. In quaternion notation, a proper (i.e., non-inverting) rotation in SO(4) is a proper simple rotation if and only if the real parts of the unit quaternions Q L and Q R are equal in magnitude and have the same ...