What shape do we get when we shear an ellipse? And more generally, do affine transformations always map conic sections to conic sections?
What shape do we get when we shear an ellipse? Is it another ellipse (or circle in special cases)? Or is it some other shape which isn’t a conic section?
I was under the impression that applying any affine transformations to an ellipse would always yield an ellipse (or special ellipses like circles, line segments or points).
Yet recently, I was doing an exercise in my Linear Algebra textbook, and in one of the answers it states that:
“An ellipse has the form $Ax^2 + By^2 = C$ where $A ≠ B$.”
Well clearly, I disagree with the condition A ≠ B for a start, as I implied above that I consider circles to be a subset of ellipses.
Secondly, a rotated ellipse will not be of that form, either. This leads me to believe that this is a mistake in the book (the question/answers do not say that the ellipse has to have major and minor axis in the same orientation as the $x$/$y$ axes either).
Is it, or am I missing something here?
This textbook I’m using is the Further Maths HL textbook for IB, and although 1st edition, all of the content in all of the IB maths textbooks so far is very good, and the answers have always been (until now) trustworthy.
This led me to a few other speculations as well… so in addition to my first question:
- What shape do we get when an ellipse is sheared?
Could someone clarify these following ones too?
- When ellipses are affine transformed, is the image always an ellipse?
- When parabolas/hyperbolas are affine transformed, are the images always parabolas/hyperbolas?
- And hence, from the above 2 questions, does the affine transformation of a conic section always result in a conic section?
The question in the book (although not necessarily relevant, as I am just challenging the statement above in the answers) is:
Under what conditions does a linear transformation with matrix \begin{bmatrix} a&b\\ c&d \end{bmatrix} transform the unit circle into:
a. an ellipse? b. a circle?
Over on the Wikipedia page on conic sections you can see that we can write a general conic section as: $$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$$ And this can be rewritten as an equation: $$X^TMX + c=0$$ Where $$X = \left[ \begin{matrix}x\\y\\\end{matrix} \right]$$ $$M = \left[ \begin{matrix}A&B/2\\B/2&C\\\end{matrix} \right]$$ and $c$ contains the linear terms.
When we do a general linear transformation ($X\to TX$ with $T$ invertible) of the plane (rotation, shear, or scale) this equation transforms to: $$(TX)^TMTX + c'=0$$ $$XT^TMTX + c'=0$$ So $M\to T^TMT$ is the transformation of the coefficients of the conic section.
Then, since the sign of the discriminant $B^2-4AC = -4\det M$ determines the type of conic section, and the transformation $\det M \to (\det T)^2\det M$ preserves the sign, all linear and affine transformations of the plane map conics to conics of the same type (ellipses to ellipses, parabolas to parabolas, and hyperbolas to hyperbolas). But only orthogonal transformations preserve circles, of course.
Yes; affine transformations map conic sections to conic sections.
The correct statement that your text should have made is
"An axis-aligned ellipse, centered at the origin, has the equation .... When $A = B$, the ellipse is also called a circle.