8

I came across the picture below through random means.

ellipse magic

What is being demonstrated? All I could think of is maybe the center of the triangle is moving back and forth between the focii of the ellipse, but even if that's true (which it may or may not be - it's purely conjecture on my part), there's clearly more going on here.

Ben Thul
  • 195
  • 1
    It seems that the side lengths of the triangle are fixed, and two of the vertices are each constrained to be on one of the two lines. The resulting locus of the third vertex is an ellipse. I can't think of a deeper explanation right now. – JimmyK4542 Jun 27 '15 at 04:59
  • 1
    The origin of this appears to be a blog post about the "Van Schooten Ellipse", which has some discussion and links to other ellipse-drawing mechanisms. – Blue Jun 27 '15 at 06:14
  • I foresee a bunch of calculus/analytic geometry students doing battle with this some time in the future. – Jyrki Lahtonen Jun 27 '15 at 12:07
  • The points moving along the diameters are the feet of perpendiculars to those diameters from a point moving along the circumference. – Blue Jun 27 '15 at 13:37

2 Answers2

3

We need to prove the following: when two vertices of a fixed triangle slide along two arms of a fixed angle, the locus described by the third vertex is an ellipse.

In the figure all the letters excepting $t$ and $P(x,y)$ are data of the problem; the angle $t$ determines the position of the vertex $P(x,y)$ so we choose $t$ as a parameter to be eliminated in order to find out the searched locus.

enter image description here

We have $$OB=\frac {c\sin (\theta+t)}{\sin\theta}$$ then $$x= \frac {c\sin (\theta+t)}{\sin\theta} \,–\, a\cos(\beta+t)$$ $$y=a\sin(\beta+t)$$ i.e. $$x=A\cos t-B\sin t$$

$$y=C\cos t+D\sin t$$

Where $$A=c-a\cos\beta$$$$B=c\cot \theta +a\sin\beta$$ $$C=a\sin\beta$$ $$D=a\cos\beta$$

Solving for $\sin t$ and $\cos t$ and because of $$\sin^2 t+\cos^2 t=1$$ it follows

$\det^2\begin{pmatrix}x&-B\\y&D\\ \end{pmatrix}$+$\,\det^2\begin{pmatrix}A&x\\C&y\\ \end{pmatrix}=$$\,\det^2\begin{pmatrix}A&-B\\C&D\\ \end{pmatrix}$

Thus $$(Dx+By)^2+(Ay-Cx)^2=(AD+BC)^2$$i.e.$$(C^2+D^2)x^2+(A^2+B^2)y^2+2(BD-AC)xy=(AD+BC)^2$$ This is a quadratic form of two real variables in which coefficients of $x^2$ and $y^2$ are both positive and distinct. This shows that the locus is an ellipse.

Piquito
  • 29,594
  • Not necessarily distinct. If $\theta = \frac\pi 2$ (the two lines are perpendicular) and $\cos\beta=\frac c{2a}$ (the triangle is isosceles) then $A^2+B^2=C^2+D^2$ and the ellipse may become a circle (but the triangle would probably have to have a zero height for that). – CiaPan Jun 28 '15 at 23:29
  • @CiaPan:When $\theta = \frac\pi 2$ the triangle $\Delta ABC$ have no restriction to $\alpha=\beta$; you can choose it as you want (inside the reasonable of course and, if I am not wrong, this case of $\theta = \frac\pi 2$ was the only one studied by Van Schooten in the 17th century) – Piquito Jun 29 '15 at 00:00
  • Of course we're not limited to isosceles triangles. However, if you choose $\alpha\approx 0$ and $\beta \gg 0$, that is a $C$ vertex close to $B$, then you get an ellipse close to the horizontal axis (degenerated to a line segment if $C=B$). I know the resulting curve is a circle when $C$ is a middle of the $AB$ segment—and I suspect, it's only then. – CiaPan Jun 29 '15 at 00:14
  • @CiaPan: I wanted to answer this way but the resulting equation was difficult to simplify: $$x=\sqrt {A^2+B^2}(\sin\theta_1 cost-\cos\theta_1 sint)=\sin(\theta_1-t)$$ Similarly $$y=\sqrt{C^2+D^2}\sin(\theta_2+t)$$ i.e.$$\theta_1-t=arc\sin \frac{x}{\sqrt{A^2+B^2}}$$ $$\theta_2+t= arc\sin \frac{y}{\sqrt{C^2+D^2}}$$ then $$\theta_1+\theta_2=arc\sin \frac{x}{\sqrt{A^2+B^2}}+arc\sin \frac{y}{\sqrt{C^2+D^2}}$$ – Piquito Jun 29 '15 at 01:19
1

You are given a circle with two diameters that are not perpendicular to each other. Two points go back and forth along the diameters. A third point creates a triangle. As the two points move along the diameters, the third point of the triangle traces an ellipse.

Robert
  • 91
  • Will that always be true? That is given any two non-perpendicular diameters and any third point, will it always create an ellipse? – Ben Thul Jun 27 '15 at 12:37
  • Sure. Taking perpendicular lines will give ellipse, too (possibly a special case of ellipse, namely a circle, if a triangle was isosceles). – CiaPan Jun 28 '15 at 23:20