Quadratic Curve

Author

Eric W. Weisstein
March 16, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneCurves/QuadraticCurve.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/QuadraticCurve.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Equations

trans=Thread[{x,y}->{{Cos[θ],Sin[θ]},{-Sin[θ],Cos[θ]}}.{x',y'}];

{x,y,x y,x^2,y^2}/.trans//ColumnForm

(new=a x^2+2b x y+c y^2+2d x+2f y+g/.trans)//TraditionalForm

Collect[Expand[new],{x'^2,x',x'y',y',y'^2}]//TraditionalForm

Transformation

With[
{
a=(Subscript[t, 1])((Cos[θ])^2)+(Subscript[t, 2])((Sin[θ])^2),
c=(Subscript[t, 1])((Sin[θ])^2)+(Subscript[t, 2])((Cos[θ])^2)
},a-c]

FullSimplify[%]