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[parent] singular points of plane curve (Topic)

The points of a plane curve

$\displaystyle x = x(t), \quad y = y(t)$
in which both derivatives $ x'(t)$ and $ y'(t)$ vanish, are in general singular points of this curve. For studying such points we suppose that
$\displaystyle x'(t_0) = y'(t_0) = 0$
and that $ x(t)$ and $ y(t)$ have in a neighbourhood of $ t_0$ the derivatives of all orders. Thus we have the Taylor expansions
\begin{align*}\begin{cases}x(t) = x_0+a_2(t-t_0)^2+a_3(t-t_0)^3+\ldots\\ y(t) = y_0+b_2(t-t_0)^2+b_3(t-t_0)^3+\ldots, \end{cases}\end{align*} (1)

where $ x_0 = x(t_0),\;\, y_0 = y(t_0)$.

Assume now that $ a_2$ and $ b_2$ are not both 0. Then the slope of the chord (secant line) between the points $ (x_0,\,y_0)$ and $ (x,\,y)$ is $ m = (y-y_0)\!:\!(x-x_0)$ and by (1) its limit as $ t \to t_0$ equals $ \frac{b_2}{a_2}$ (or the limit of $ 1/m$ is $ \frac{a_2}{b_2}$). Accordingly, the curve has in the point $ (x_0,\,y_0)$ a definite tangent line (which may be vertical if $ a_2 = 0$). From the expression

$\displaystyle y-y_0 = b_2(t-t_0)^2+b_3(t-t_0)^3+\ldots$
one sees that when $ \vert t-t_0\vert$ is sufficiently small, the difference $ y-y_0$ of the ordinates has the same sign as $ b_2$, i.e. the sign is the same on both sides of $ t_0$. This means that the curve has a cusp at the point.

If the slope angle of the tangent line is $ \alpha$, we have

$\displaystyle \sin\alpha = \frac{b_2}{\sqrt{a_2^2+b_2^2}}, \quad \cos\alpha = \frac{a_2}{\sqrt{a_2^2+b_2^2}}.$
We can form the projection of the chord to the normal line of the tangent, obtaining
$\displaystyle (x-x_0)\cos(\alpha+\frac{\pi}{2})+(y-y_0)\sin(\alpha+\frac{\pi}{2... ...\!-\!t_0)^3\!+\!(a_2b_4\!-\!a_4b_2)(t\!-\!t_0)^4\!+\ldots}{\sqrt{a_2^2+b_2^2}}.$ (2)

  • The case $ a_2b_3-a_3b_2 \neq 0$. When $ \vert t-t_0\vert$ is sufficiently small, the expression (2) of the projection changes its sign at the same time as $ t-t_0$ (due to the third power). Thus the two branches of the curve are on different sides of the tangent line. One speaks of a ordinary cusp.
  • The case $ a_2b_3-a_3b_2 = 0$ but $ a_2b_4-a_4b_2 \neq 0$. The expansion (2) begins with the term with the even power $ (t-t_0)^4$, the projection keeps its sign as $ t$ passes through $ t_0$. Therefore the both branches are on the same side of the tangent line. Now there is a ramphoid cusp (in German die Schnabelspitze) on the curve.

Example. Examine the singular points of the algebraic curve

$\displaystyle (y-x^2)^2 = x^5.$
Let us take the ratio $ \displaystyle\frac{y}{x^2} := t$ as the parametre. This yields first $ (x^2t-x^2)^2 = x^5$; dividing by $ x^4$ gives the parametric presentation
\begin{align*}\begin{cases}x = (t-1)^2,\\ y = (t-1)^4t. \end{cases}\end{align*}    

The derivatives $ \frac{dx}{dt} = 2(t-1)$ and $ \frac{dy}{dt} = 4(t-1)^3t+(t-1)^4$ have the common zero $ t = 1$, whence there is a cusp in the point $ (0,\,0)$. Now the Taylor expansions in $ t = 1$ are the polynomials
\begin{align*}\begin{cases}x = (t-1)^2,\\ y = (t-1)^4(1+(t-1)) = (t-1)^4+(t-1)^5. \end{cases}\end{align*}    

Thus $ a_2 = b_4 = b_5 =1,\;\, a_3 = a_4 = b_2 = b_3 =0$, and accordingly $ a_2b_3-a_3b_2 = 0$, $ a_2b_4-a_4b_2 = 1 \neq 0$. It is a question of a ramphoid cusp. Both branches start from the origin to the right, their common tangent is the $ x$-axis. Note that the curve may be given in the form $ y = x^2(1\pm\sqrt{x})$.
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"singular points of plane curve" is owned by pahio.
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See Also: cusp

Also defines:  ordinary cusp, ramphoid cusp

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Cross-references: origin, polynomials, parametric presentation, parametre, ratio, passes through, even power, term, branches, third power, tangent, normal line, projection, slope angle, cusp, ordinates, difference, expression, tangent line, limit, secant line, chord, slope, Taylor expansions, neighbourhood, curve, singular points, vanish, derivatives, plane curve, points
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This is version 7 of singular points of plane curve, born on 2008-03-27, modified 2008-03-28.
Object id is 10448, canonical name is SingularPointsOfPlaneCurve.
Accessed 544 times total.

Classification:
AMS MSC: 51N05 (Geometry :: Analytic and descriptive geometry :: Descriptive geometry)
 53A04 (Differential geometry :: Classical differential geometry :: Curves in Euclidean space)
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