in some coordinates. In my post on the capricornoid, I gave some old-fashioned names for different kinds of cusps, and in a comment I mentioned that the cusp of order 5/2 was called a **rhamphoid cusp**. Strangely, I wrote all that *before* knowing that Arnol’d places great significance on the cusp of order 5/2 in the involute of a cubical parabola!

]]>

For anyone with Mathematica who wants to test this: the image produced by the code below shows a tangent to a circle, in red, passing through the point of tangency, but the circle itself, in blue, which ought to pass through the same point, does not.

theta = Pi/8;

point = {Cos[theta], Sin[theta]};

tangent = {-Sin[theta], Cos[theta]};

width = height = 1/100;

Graphics[

{

{Blue,Circle[{0,0},1]},

{Red,Line[{point-tangent,point+tangent}]},

{Black,PointSize[.02],Point[point]}

},

PlotRange->{

{point[[1]]-width/2,point[[1]]+width/2},

{point[[2]]-height/2,point[[2]]+height/2}}

]

On the question of the “best” rendering of a continuous curve, I suppose the gold standard would be something like a perfectly anti-aliased image of a finite-width region whose borders lie at fixed orthogonal distances on either side of the idealised zero-width curve, with some well-defined “capping” behaviour at any endpoints and/or points of discontinuous tangent. In reality, a lot of systems do a pretty good job at this for straight line segments, Bezier curves, and (sometimes) circular arcs, but anything else has to be rendered by combining those elements.

]]>I would like to do something with the capricornoid on my blog *Visual Insight*, but a simple static image of this curve just isn’t charismatic enough for that blog. Varying the parameters allows for a nice explanation of what can happen to a tacnode as a curve changes.

form, i.e. the sum of the terms with highest degree in a

polynomial. To avoid negative signs we can write the

polynomial as

Suppose there are quadratic polynomials , such

that

,

is an equation for one of the ellipses, and

is an equation for the other ellipse.

The degree form for is the product of the degree forms

for and . After expanding we get its

degree form

The degree form for is a real quadratic form

with positive discriminant, . The ellipses would

have to be mapped to each other by the reflection

so the degree form for is

The product of the degree forms for and is

By equating corresponding coefficients in and

we get and .

Since the discriminant is positive and must have the

same sign. So and

This contradicts . Therefore the polynomials

, do not exist.

But anyway, I eventually figured it out, and I’m realy glad that this free service exists! If I ever do more math programming, I’ll consider paying the fee for extra power.

]]>