What happens if a flat geometric shape, such as a rectangle, begins to rotate rapidly relative to one of its sides?
We create a new geometric solid in space by rotation.
The side surfaces of the cylinder are formed by the sides of a rotating rectangle.
The official definition for such geometric bodies is as follows:
The solids of revolution are volume bodies that arise when a flat geometric figure bounded by a curve rotates around an axis lying in the same plane.
Here it is important that a flat geometric figure can be a completely arbitrary shape.
For example, a curve that, when rotated, forms a vase or a light bulb. Such tools for creating bodies of revolution are very popular with those who work in 3D-design programs.
But from a mathematical point of view, the following geometric bodies of revolution are of primarily interest to us:
The cylinder is formed by a rectangle rotating around one of the sides.
The cone is formed by a rectangular triangle rotating around one of the legs.
A truncated cone is part of the cone located between its base and the cutting plane parallel to the base.
It is formed by rotating a rectangular trapezoid around its side, perpendicular to the bases of the trapezoid.
The ball is formed by a semicircle rotating around the diameter of the cut.
During the rotation of the contours of the figures, a surface of revolution arises (for example, a sphere formed by a circle), while during the rotation of filled contours, bodies appear (like a ball formed by a circle).
An ellipsoid is a surface in three-dimensional space, obtained by deforming a sphere along three mutually perpendicular axes.
The torus is formed by a circle rotating around a straight line that does not intersect it.
The usual sense of the torus is a "bagel".
A paraboloid is a surface that is formed as a result of a rotation around the axis of a curve formed by a graph of a parabola. Hence the name parabol-oid.
A hyperboloid is a surface that is formed as a result of rotation around the axis of a curve formed by a graph of a hyperbola. Accordingly, the name is hyperb-o-loid.
In order to compare the sizes of the resulting models, we tried to assemble them on the same surface together with prisms from the Magic Edges #16 issue.
It turned into a whole mathematical city of paper that fits on the table!