Plane curve

A plane curve is the zero set of a polynomial in two variables, or a one-dimensional curve embedded in a plane.

An astroid is the four-cusped plane curve parametrized by $x=a{\cos }^{3}t$, $y=a{\sin }^{3}t$ for $a>0$. It appears as the relative trajectory in the astroid motion of a harmonic oscillator in a rotating frame. The parametrization has zero first derivative at the cusps while remaining smooth as a function of its parameter.

This irreducible plane branch has parametrization $x=t^5$, $y=t^2$. Its only singular point is the origin. The resolution of the (2,5) cusp by two blowups makes its strict transform smooth in every characteristic.

Blow up the origin and use the chart $x=uy$, giving the strict transform $u^2-y^3=0$. Blow up its remaining singular point and use $u=vy$, giving $v^2-y=0$, which is smooth because its derivative in $y$ is $-1$. The complementary chart at each stage has no point of the strict transform on its exceptional divisor. These two blowups of a smooth algebraic surface resolve the branch, although the smooth transform is still tangent to an exceptional divisor.

An Archimedean spiral has radius linear in polar angle, $r=a+b\theta$. For the special parametrization $(u\cos u,u\sin u)$, the curvature of a plane curve is $(u^2+2)/(1+u^2{)}^{3/2}$. Its parameter speed is $\sqrt{1+u^2}$, so it remains a regular curve at the origin.

A superelliptic curve has an affine equation $y^m=f(x)$. When $f$ is square-free and the characteristic does not divide $m$, projection to the $x$-coordinate is a finite morphism whose finite ramification occurs over the roots of $f$; its behavior at infinity depends on $m$ and $\mathrm{deg}f$.

For a smooth plane curve of degree $d\ge 4$, adjunction givesIts canonical map is induced by the $(d-3)$rd Veronese map and is therefore an embedding.

A semicubical parabola is a cusped cubic curve affinely equivalent to $y^2=x^3$.

The homomorphismhas kernel $(y^2-x^3)$ and image $k[t^2,t^3]$. The quotient is therefore an integral domain, so the curve is irreducible. In characteristic zero its gradient $(-3x^2,2y)$ vanishes at the origin, making the origin its unique singular point of an algebraic variety.