Spiral similarity
Contents
[hide]Basic information
A spiral similarity is a plane transformation composed of a rotation of the plane and a dilation of the plane having the common center. The order in which the composition is taken is not important.
The transformation is linear and transforms any given object into an object homothetic to given.
On the complex plane, any spiral similarity can be expressed in the form where
is a complex number. The magnitude
is the dilation factor of the spiral similarity, and the argument
is the angle of rotation.
The spiral similarity is uniquely defined by the images of two distinct points. It is easy to show using the complex plane.
Let with corresponding complex numbers
and
so
Case 1 Any line segment can be mapped into any other
using the spiral similarity. Notation is shown on the diagram.
is circle
is circle
is any point of
is circle
is the image
under spiral symilarity centered at
is the dilation factor,
is the angle of rotation.
Case 2 Any line segment can be mapped into any other
using the spiral similarity. Notation is shown on the diagram.
is circle
(so circle is tangent to
is circle tangent to
is any point of
is circle
is the image
under spiral symilarity centered at
is the dilation factor,
is the angle of rotation.
Simple problems
Explicit spiral symilarity
Given two similar right triangles and
Find
and
Solution
The spiral symilarity centered at with coefficient
and the angle of rotation
maps point
to point
and point
to point
Therefore this symilarity maps to
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Hidden spiral symilarity
Let be an isosceles right triangle
Let
be a point on a circle with diameter
The line
is symmetrical to
with respect to
and intersects
at
Prove that
Proof
Denote
Let
cross perpendicular to
in point
at point
Then
Points and
are simmetric with respect
so
The spiral symilarity centered at with coefficient
and the angle of rotation
maps
to
and
to point
such that
Therefore
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Linearity of the spiral symilarity
Points
are outside
Prove that the centroids of triangles and
are coinsite.
Proof
Let where
be the spiral similarity with
and
A vector has two parameters, modulo and direction. It is not tied to a center of the spiral similarity. Therefore
We use the property of linearity and get
Let
be the centroid of
so
is the centroid of the
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