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Reflection in a Point



A point P'(x',y') is said to be the reflection of a point P(x,y) in the point C(H,K) if C is the midpoint of the segment P P'. Using this definition show the following.

A. The transformation equations for a reflection in a point are

x'=2H-x and x=2H-x'

y'=2K-y and y=2K-y';

B. The reflection of the line a x+b y+c=0 in the point (H,K) is

a x + b y-(2a H+2b K+c)=0;

C. The reflection of the quadratic "reflctpt_1.gif" in the point (H,K) is



Also, verify that the reflection in a point transformation is equivalent to a rotation of π radians about the reflection point (H,K).


Solve the midpoint relationship for the coordinates of the transformation. Substitute the reflected coordinates into the equation of a line to produce a reflected line. Substitute the reflected coordinates into the equation of a quadratic to produce the reflected quadratic. Apply the proposed rotation to show it is equivalent to the reflection.


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(H,K) is the midpoint of PP'. Solve for (x,y) and (x',y').  This is the solution to proposition A.



Reflect a line through a point.  This is the solution to proposition B.

eq1=a*x+b*y+c /. {x->2H-x,y->2K-y};


Reflect a quadratic through a point.  This is the solution to proposition C.

eq2=a*x^2+b*x*y+c*y^2+d*x+e*y+f /. {x->2H-x,y->2K-y};


The reflection is the same as the specified rotation.  This is the solution to the final proposition.



Copyright © 1999-2007 Donald L. Vossler, Descarta2D Publishing