Curve

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

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 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 helix is a space curve that winds around an axis while advancing along it at a constant rate. A circular helix can be parametrized by $(R\cos t,R\sin t,at)$.

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.

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.