Tuesday, August 18, 2015

Unizor - Geometry3D - 3D Similarity

Unizor - Creative Minds through Art of Mathematics - Math4Teens

3-D Similarity

The definition of scaling or homothety in three-dimensional space is exactly the same as on a plane. The transformation of scaling requires the center of scaling (a point in a space) and the factor of scaling (non-zero real number) with exactly the same functionality as on a plane.

Assume, point X is a center of scaling and f is a factor of scaling. Any point M in a space not coinciding with the center X is transformed by a scaling using the following procedure (exactly as on a plane):
(a) connect points M (given) and X (center) by a segment;
(b) construct a new segment with the length equal to the length of XM multiplied by an absolute value of the factor |f|;
(c) starting at the center X along the line XM position this new segment in the direction from X to M for positive factor f or to an opposite direction for negative f;
(d) the end of this new segment is the result of transformation of point M by scaling with center X and factor f.
If point M coincides with center X, scaling with any factor does not change its position since the length of a segment XM is zero and will be zero after multiplication by any factor.

The most important property of scaling with factor f is the fact of changing of the length of any segment MN into another M'N' with the length that is equal to a product of the length of the original segment and factor |f|, regardless of a center of scaling X. This simple fact follows from the similarity on a plane of triangles ΔXMN and ΔXM'N'.

Now, with scaling transformation defined, we can define similarity of two geometrical objects ξ and η in three-dimensional space as the possibility to transform one into another using scaling, as defined above, optionally combined with congruent transformations of parallel shift (translation), rotation and symmetry. The word "possibility" means here the existence of a particular point X in space that can be used as a center of scaling and a particular non-zero factor of scaling f that, if used in the scaling of object ξ, transform this object into another, congruent to object η.

The three-dimensional expansion of this statement is that the scaling of a plane in three-dimensional space is a plane parallel to the original. To prove it, it's sufficient to choose any two intersecting lines on the original plane and compare them with their counterparts on the resulting plane. Since these lines are correspondingly parallel, the planes they belong to are parallel as well.

Dihedral angles are also preserved by scaling since they are measured by angles between two lines, correspondingly belonging to two planes and perpendicular to a line of their intersection, and the angles between lines are preserved.

It's easy to prove (admittedly, not absolutely rigorously since we do not use the full power of the theory of limits) that scaling of any polygon by a factor f effects its area by a factor f².

It can also be proven (admittedly, not absolutely rigorously since we did not use the full power of the theory of limits) that scaling of any polyhedron by a factor f effects its volume by a factor |f|³.

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