Homothety compositions
by math_explorer, Feb 5, 2012, 1:31 AM
位似 
旋轉 = 位似旋轉
homothety rotation = spiral similarity
*sigh*
Theorem. A plane transformation takes every pair of points
to a pair 
with 
for some fixed constant 
iff it is a homothety.
pf. (if) Similar triangles.
(only if) Send a point through the transformation, pick out the hypothetical center with ratios, and verify the rest of the plane.
zzzzzzzzz
Given two points
, any homothety on line that does not map 
to 
can be uniquely represented as the composition of a homothety centered at followed by a homothety centered at .
pf. Pick two random points
not on line so that the homothety we're trying to construct maps 
to 
. From the fact that it doesn't map to we can deduce that 
and 
are not parallel, so they intersect, say at point 
. Take the homothety centered at mapping to followed by the homothety centered at mapping to . The resulting transformation is a homothety (first theorem three times), which is centered on line by symmetry. Its center must also be on line 
; the intersection exists and is unique; it maps to and the ratio is also unique.
Yeah, today I'm in a hand-waving mood.
旋轉 = 位似旋轉homothety
rotation = spiral similarity*sigh*
Theorem. A plane transformation takes every pair of points
to a pair 
with 
for some fixed constant 
iff it is a homothety.pf. (if) Similar triangles.
(only if) Send a point through the transformation, pick out the hypothetical center with ratios, and verify the rest of the plane.
zzzzzzzzz
Given two points
, any homothety on line that does not map 
to 
can be uniquely represented as the composition of a homothety centered at followed by a homothety centered at .pf. Pick two random points
not on line so that the homothety we're trying to construct maps 
to 
. From the fact that it doesn't map to we can deduce that 
and 
are not parallel, so they intersect, say at point 
. Take the homothety centered at mapping to followed by the homothety centered at mapping to . The resulting transformation is a homothety (first theorem three times), which is centered on line by symmetry. Its center must also be on line 
; the intersection exists and is unique; it maps to and the ratio is also unique.Yeah, today I'm in a hand-waving mood.
May 2020
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