Tuesday, October 20, 2015

Unizor - Geometry3D - Similarity of Cylinders, Cones, Spheres

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Notes to a video lecture on http://www.unizor.com

Similarity of Cylinders, Cones and Spheres

We have discussed the fact that scaling of a straight line transforms it into a straight line and scaling of a plane transforms it into a plane.
We have also discussed that angles between lines and dihedral angles between planes are preserved by scaling.

Based on this, we can say that any polyhedron is transformed by scaling into a polyhedron with the same number of vertices, edges and faces, with the same angles between edges and with the same dihedral angles between faces. So, the general shape of a polyhedron is preserved.

In this topic we will attempt to prove that the scaling of other geometrical objects - cylinders, cones and spheres - preserves their type.

Cylindrical Surface

Cylindrical surface is characterized by its directrix and generatrix. In particular, we draw a line parallel to generatrix through each point on a directrix.
Now let's scale each such line. A transformed image of this line by a scaling will be, as we know, a straight line parallel to an original. Therefore, no matter how a directrix is transformed, the new object will still be a cylindrical surface because it will consists of straight lines parallel to the same generatrix.

Right Circular Cylinder

Recall that all angles are preserved by scaling. In a right circular cylinder any line forming its side surface is parallel to a generatrix, which, in turn, is perpendicular to a plane that contains a circular directrix and both base planes. In the image of this right circular cylinder all such side lines will be still parallel to an original generatrix and an image of a plane that contains a circular directrix and images of both bases would be corresponding parallel planes. Therefore, perpendicularity between a generatrix of a cylinder and a plane where its directrix lies is preserved. Therefore, scaling transforms a right circular cylinder into a right circular cylinder.

Right Circular Cone
A right circular cone is the one whose altitude (a perpendicular form an apex onto a circular base) is falling into a base's center.
Let point S be an apex of an source right circular cone, plane β - its base plane, point O - a center of its circular base and points A and B - two points on a base circle.
Since this is a right circular cone, SO⊥β, SO⊥OA and SO⊥OB.
Let S', β', O', A' and B' be images of corresponding points after scaling.
Since all angles are preserved, S'O'⊥O'A' and S'O'⊥O'B'. Therefore, S'O'⊥β', that is, S'O' is an altitude of a cone's image. Hence, an image of a right circular cone is a right circular cone.

Sphere

Since equality of the lengths of two segments is preserved by scaling, images of two points on a sphere, equidistant from its center by definition of a sphere, will be equidistant from an image of a center. Therefore, an image of a sphere after scaling is a sphere.