Graph of a Relation under a Linear Coordinate Transformation

graph of relation P

square

disk

triangle

a

0.5

b

4

c

0.5

d

6.4

move

(X, Y)

(x, y)

R(X, Y) = False

P(x, y) = False

Suppose a graph of a relation

P

in two variables is given, and a relation

R

is defined by

R(X,Y)⇔P(a(X-b),c(Y-d))

.

Then the point

(X,Y)

is on the graph of

R

iff the point

(x,y)=(a(X-b),c(Y-d))

is on the graph of

P

, and

(X,Y)=(x/a+b,y/c+d)=(x/a,y/c)+(b,d)

.

So to get the graph of

R

, the graph of

P

is scaled by

(1/a,1/c)

and translated by

(b,d)

.