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Reflection in a Point

reflctpt.html

Exploration

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

"reflctpt_2.gif"

"reflctpt_3.gif"

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

Approach

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.

Initialize

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<<Descarta2D`

Solution

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

Clear[x,y,x1,y1,H,K];
{{Solve[(x+x1)/2==H,x1],
  Solve[(y+y1)/2==K,y1]},
{Solve[(x+x1)/2==H,x],
  Solve[(y+y1)/2==K,y]}}

"reflctpt_4.gif"

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

Clear[a,b,c];
eq1=a*x+b*y+c /. {x->2H-x,y->2K-y};
Map[Times[-1,#]&,Line2D[eq1,{x,y}]]

"reflctpt_5.gif"

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

Clear[d,e,f];
eq2=a*x^2+b*x*y+c*y^2+d*x+e*y+f /. {x->2H-x,y->2K-y};
Quadratic2D[eq2,{x,y}]

"reflctpt_6.gif"

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

Rotate2D[{x,y},Pi,{H,K}]

"reflctpt_7.gif"


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