Difference between revisions of "Spiral similarity"
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On the complex plane, any spiral similarity can be expressed in the form <math>T(x) = x_0+k (x-x_0),</math> where <math>k</math> is a complex number. The magnitude <math>|k|</math> is the dilation factor of the spiral similarity, and the argument <math>\arg(k)</math> is the angle of rotation. | On the complex plane, any spiral similarity can be expressed in the form <math>T(x) = x_0+k (x-x_0),</math> where <math>k</math> is a complex number. The magnitude <math>|k|</math> is the dilation factor of the spiral similarity, and the argument <math>\arg(k)</math> is the angle of rotation. | ||
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| + | The spiral similarity is uniquely defined by the images of two distinct points. It is easy to show using the complex plane. | ||
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| + | Let <math>A' = T(A), B' = T(B),</math> with corresponding complex numbers <math>a', a, b',</math> and <math>b,</math> so | ||
| + | <cmath>a' = T(a) = x_0 + k (a - x_0), b' = T(b) = x_0+ k (b-x_0) \implies</cmath> | ||
| + | <cmath>k = \frac {T(b) - T(a)}{b-a} = \frac {b' - a' }{b - a},</cmath> | ||
| + | <cmath>x_0=\frac {ab' - ba' }{a-a'+b' -b}, a' - a \ne b' - b.</cmath> | ||
Revision as of 02:21, 10 June 2023
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
![\[a' = T(a) = x_0 + k (a - x_0), b' = T(b) = x_0+ k (b-x_0) \implies\]](http://latex.artofproblemsolving.com/d/c/4/dc431dd6b50b0a6854acd32c8b0703a866714819.png)
![\[k = \frac {T(b) - T(a)}{b-a} = \frac {b' - a' }{b - a},\]](http://latex.artofproblemsolving.com/7/6/7/76710e0fd696d8c5b9cc0b46c37034a89fed60d0.png)
![\[x_0=\frac {ab' - ba' }{a-a'+b' -b}, a' - a \ne b' - b.\]](http://latex.artofproblemsolving.com/5/9/2/592a33728cc82f7919a8d6bca388b3a46cd497b4.png)
