The Space of Inner Products

quadratic form parameters

a

0.5

b

0.5

δ

0.2

group parameters

α

1

β

0

γ

0

show orbit

This Demonstration shows the space of triples

(a,b,c)

with

ab>

2

c

, which can be identified with the positive definite quadratic form

a

2

x

+cxy+b

2

y

, together with the orbits of the natural action of the special linear group

SL(2,)

of

2×2

matrices with determinant 1. You can vary the quadratic form by changing its three parameters (

a

,

b

, and the discriminant

δ=ab-

2

c

) and see the point corresponding to the form move within the region of space where the positive definite quadratic forms lie. By varying the matrix parameters you can see the image of the fixed form (which corresponds to another quadratic form with the same discriminant). You also see the part of the orbit of the action contained within the displayed area by checking the "show orbit" checkbox.