Curve

A curve in a topological space $X$ is a continuous function from an interval into $X$.

An intersection curve is a one-dimensional intersection of geometric objects, commonly two surfaces in three-dimensional Euclidean space. If the surfaces are regular level sets $F=0$ and $G=0$ and their gradients are linearly independent at a common point, the implicit function theorem makes their intersection locally a smooth curve. Its tangent vector is parallel to $\nabla F\times \nabla G$. The sphere-plane intersection gives a circle in the transverse case; tangency can instead collapse the intersection to a point, so the dimension cannot be assumed without a regularity condition.

A plane curve is a curve contained in a plane. In algebraic geometry, one studies plane curves defined by polynomial equations. General plane curves, such as an Archimedean spiral, need not be algebraic curves.

A tractrix is a plane curve whose tangent segment from the point of contact to a fixed axis has constant length. For segment length one and a positive acute tangent angle $\theta$, its height is $r=\sin \theta$. The slope relation $dr/dh=\tan \theta$ gives $dh/d\theta ={\cos }^2\theta /\sin \theta$. Integrating yields the displayed axial coordinate up to an additive constant. Conversely these coordinates have the required slope, and their tangent intercept is at distance $r/\sin \theta =1$. Rotating this profile about its axis gives a tractoid.

The trajectory of a point at fixed distance $a$ from the centre of a circle of radius $r$ rolling along a line. In coordinates aligned with that line it has a linear centre displacement plus sinusoidal components. The case $a=r$ is a cycloid; $a<r$ gives a shortened trajectory and $a>r$ gives loops. An inclined rolling line rotates this geometry relative to horizontal and vertical coordinates.

A point at distance $a$ from the axis of a radius-$r$ cylinder rolling downhill at constant axis speed $v$ has $\dot{x}=v[\cos \alpha +(a/r)\cos \theta ]$, $\dot{y}=-v[\sin \alpha +(a/r)\sin \theta ]$. For $0\le \alpha <\pi /2$, the existence of $\dot{x}=0$, $\dot{y}>0$ is equivalent to $a>r$. At $a=r$ the relevant zero-horizontal-velocity point is stationary, so a point within or on the rolling cylinder never has strictly vertical upward motion.

A hyperbolic spiral is a plane curve whose radial distance in polar coordinates is inversely proportional to the angular displacement from a fixed angle: $r=a/(\theta -{\theta }_0)$, on a branch where $r>0$. For $a>0$ and $\theta >{\theta }_0$, it winds infinitely often towards the origin as $\theta \to \infty$. Its geometric winding alone does not determine whether a moving particle traverses it in finite time: that also requires a law for the angular velocity.

An involute is the curve traced by the end of a taut string as it is unwound from a fixed curve. For a circle of radius $R$, one parametrization is $\mathbf{x}(\theta )=R(\cos \theta +\theta \sin \theta ,\sin \theta -\theta \cos \theta )$. The straight free length is $R\theta$, and differentiation shows that the traced velocity is perpendicular to the taut string. Reversing the motion winds the string back around the circle.

A point mass attached to a light inextensible string winds around a fixed smooth circular pole of radius $R$. Let the initial straight free length be $L$, and assume the string remains taut without slipping over the part already laid on the pole. Since its tension is perpendicular to the point mass's velocity, the speed $v$ stays constant. If the contact point turns through $d\theta$, the free length $s$ obeys $ds=-Rd\theta$, while the involute geometry gives $v=s\dot{\theta }$. Thus $s\dot{s}=-Rv$, or $s^2=L^2-2Rvt$. First contact of the point mass with the pole occurs at $s=0$, after $L^2/(2Rv)$. In the ideal model, the required tension $m{v}^2/s$ diverges at contact; finite string strength or finite particle size limits that idealization.

A figure-eight plane curve with two congruent lobes and a crossing at the origin. With scale $a>0$, its polar equation is $r^2=a^2\cos 2\theta$. One quarter has coordinates $x=at\sqrt{(1+t^2)/2}$, $y=at\sqrt{(1-t^2)/2}$, $0\le t\le 1$. Differentiation gives arc element $adt/\sqrt{1-t^4}$. The total arc length is therefore $4a{\int }_0^{1}dt/\sqrt{1-t^4}$.

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.

Let an Archimedean spiral have polar radius $r=at$ and angle $t$ in polar coordinates, with $0\le t\le R/a$, where $R>0$ is fixed. Parameterizing by $r$ gives the arc lengthThe bounds $r\le \sqrt{a^2+r^2}\le r+a$ yieldThe squeeze theorem proves the displayed limit. Thus the arc length grows like $R^2/(2a)$ as the turns become closely spaced, without requiring an explicit antiderivative.

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.

The Euler spiral is the planar curve parametrized by the Fresnel integrals. In the optical convention,The tangent has unit length and argument $\pi u^2/2$, so its signed curvature is $\pi u$, proportional to signed arc length. In Fresnel diffraction, an interval integral is the directed chord joining its endpoint parameters on the Cornu spiral. Chord length gives the interval factor's modulus, while chord direction gives its complex argument.

A curve $\gamma :[0,1]\to X$ whose endpoints agree, $\gamma (0)=\gamma (1)$. This is the same unbased notion as a loop; self-intersections are allowed.

An asymptote of an unbounded curve is a straight line whose distance from the curve tends to zero along a specified end. For a graph, a sloping asymptote $y=mx+b$ satisfies $y(x)-(mx+b)\to 0$ as $x$ tends to the relevant infinity. Factorization of the leading quadratic part locates the two asymptotes of a nondegenerate hyperbola.

A parametric curve is described by a vector-valued map $t\mapsto Q(t)$. Its derivative gives a tangent vector wherever it is nonzero.

// Target: geometry-and-topology.bigb

A Lissajous curve is a parametric curve with coordinates $x(t)=A\sin (at+\delta )$ and $y(t)=B\sin (bt)$. The frequency ratio and phase shift control its crossings and whether its trajectory closes.

A parametric curve in the plane, given by two coordinate functions $x(t),y(t)$. Its speed is $\sqrt{x^{\prime }(t{)}^2+y^{\prime }(t{)}^2}$ wherever these derivatives exist, giving its arc length by integration.

A parametric curve interface supplies its parameter domain, evaluation, and derivatives where they exist. Robust geometric algorithms also need restricted subcurves and certified range or derivative bounds on parameter intervals. Tangent, speed and curvature are derived from point and derivative enquiries when the relevant derivatives are regular. Finite point samples alone cannot certify the absence of every plane intersection of an arbitrary smooth curve.

Intersecting a parametric curve with a plane reduces to isolating zeros of a scalar function. A certified range for $F$ rejects intervals that do not contain zero. On an interval whose derivative range excludes zero, a sign-changing endpoint bracket gives exactly one root, which bisection method or a safeguarded Newton root-finding iteration refines. Other intervals are subdivided conservatively; tangencies and coplanar pieces need multiplicity or identity tests supported by the representation. An unresolved small interval is a possible contact, not a proof of an intersection. With finitely many transverse intersections and convergent enclosures, isolation terminates to any prescribed tolerance.

A cycloid is the plane curve traced by a point on a circle rolling without slipping along a straight line. A parametrization with rolling radius $R$ is $x=R(\theta -\sin \theta )$, $y=R(1-\cos \theta )$; reversing the sign of $y$ reflects the curve. Its cusps occur at integer multiples of $2\pi$. A descending cycloidal arc solves the classical brachistochrone problem.

For the cycloid $x=a(t-\sin t)$, $y=a(1-\cos t)$, $a>0$, $0\le t\le 2\pi$, the area between the arch and its baseline isIts arc length isThe positive sign of $\sin (t/2)$ inside this parameter interval removes the absolute value in the speed. Omitting the two endpoint cusps does not change either integral.

A space curve is a curve in three-dimensional Euclidean space.

A general helix is a regular space curve whose unit tangent vector makes a constant angle with a fixed axis direction $u$. In arc length parameter $s$, this means ${\gamma }^{\prime }(s)\cdot u$ is constant. Equivalently, the axial coordinate grows at a constant rate per unit arc length.

The familiar circular helix has constant distance from its axis:Its speed is $\sqrt{R^2+a^2}$, and its tangent's axial component is $a/\sqrt{R^2+a^2}$. Its signed pitch of a helix is $2\pi a$. A general helix need not have constant radius or constant axial displacement per turn: the logarithmic conical helix has a varying radius while retaining a constant tangent-axis angle.

The displayed space curve, $s>0$, lies on a cone and is parametrized by arc length. It has a constant tangent angle with its axis, while the turn spacing shrinks geometrically toward the limiting apex. Its curvature is $1/(\sqrt{2}s)$: differentiate its unit tangent with respect to $s$. Infinitely many turns accumulate at the apex with finite arc length; the apex itself is not a regular point of the curve.

The pitch is the axial displacement per complete turn. For a circular helix with $z=a\varphi +b$, the signed pitch for increasing angle is $2\pi a$, and its magnitude is $2\pi |a|$. Pitch is independent of the cylinder radius.

An inflection point of a smooth plane curve is a point where its signed curvature changes sign. At a regular inflection point the curvature vanishes.

An inflection point of the graph of a differentiable real function at which its derivative is zero. It has a horizontal tangent and a change of concavity, but need not be a local maximum or minimum. For a three-times continuously differentiable function, $f^{\prime }(a)=f^{\prime \prime }(a)=0$ and $f^{\prime \prime \prime }(a)\ne 0$ suffice: the expansion $f^{\prime \prime }(a+h)=f^{\prime \prime \prime }(a)h+o(h)$ gives opposite signs on the two sides. The elementary example is $f(x)=x^3$ at zero. The word “saddle” is sometimes used for this one-dimensional situation in elementary examination papers, but the usual saddle point of a scalar function has a multivariable meaning; do not merge the two concepts.

A logarithmic spiral has polar equation $r=A{e}^{b\theta }$. Its tangent meets each radial line at a constant angle.

The map $h$ is a homeomorphism from the real line to its image with the subspace topology: its continuous inverse is $h^{-1}(x,y)=\log \sqrt{x^2+y^2}$. Different parameters have different radii, so winding does not destroy injectivity. The origin is an accumulation point of the spiral but is not a point of this space.

A simple curve is an injective curve, apart from the permitted coincidence of its two endpoints when it is closed.

A space-filling curve is a surjective continuous map from an interval onto a region of dimension at least two.