Parametric curve ($Q(t)$)

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.