Search research articles

ABOUT JoVE

Overview Leadership Blog JoVE Help Center

AUTHORS

Publishing Process Editorial Board Scope & Policies Peer Review FAQ Submit

LIBRARIANS

Testimonials Subscriptions Access Resources Library Advisory Board FAQ

RESEARCH

JoVE Journal Methods Collections JoVE Encyclopedia of Experiments Archive

EDUCATION

JoVE Core JoVE Business JoVE Science Education JoVE Lab Manual Faculty Resource Center Faculty Site

Terms & Conditions of Use

Related Concept Videos

Calculus with Parametric Curves: Surface Areas

Calculus with Parametric Curves: Surface Areas

A parametric curve is a description of a path in the plane where both the x and y coordinates are functions of a single parameter, typically denoted t. When such a curve is revolved about an external axis lying in the same plane, it generates a surface of revolution in three dimensions. The surface area of this rotated shape depends fundamentally on two aspects: the geometry of the original curve and how far it lies from the chosen axis of rotation.A torus is a classical surface of revolution...

Parametric Surfaces

Parametric Surfaces

A parametric surface in three-dimensional space is defined through a vector-valued function\begin{equation*}\mathbf{r}(u, v) = x(u, v)\mathbf{i} + y(u, v)\mathbf{j} + z(u, v)\mathbf{k}\end{equation*}where u and v are parameters within a specified domain D in the uv-plane. The functions x(u, v), y(u, v), and z(u, v) define the coordinates of points on the surface. As u and v vary over D, the position vector r(u, v) traces a continuous surface in space. This parametric representation is essential...

Quadric Surfaces

Quadric Surfaces

Quadric surfaces are three-dimensional surfaces characterized by second-degree equations in the variables x, y, and z. These surfaces are smooth and continuous, and specific combinations of squared and linear terms define their shapes. The main types of quadric surfaces include ellipsoids, cones, paraboloids, and hyperboloids. Each type exhibits distinct geometric features depending on how the variables are arranged and related within the equation.Ellipsoids are closed surfaces formed when all...

Curve Sketching and Derivatives

Curve Sketching and Derivatives

Understanding the behavior of a function through its first and second derivatives is essential for analyzing its graph. Derivatives provide insight into where a function increases or decreases, where it attains local maxima or minima, and how its curvature behaves across different intervals.The first derivative of a function reveals the slope of the tangent line at any given point. Points where the derivative is zero or undefined are considered critical, as they often indicate potential extrema...

Tangent Planes to a Parametric Surface

Tangent Planes to a Parametric Surface

A tangent plane provides a linear approximation to a curved surface at a specific point, capturing the local behavior of the surface. It can be understood as the plane that just touches the surface at that point and is defined by the tangent directions of curves lying on the surface. These tangent directions arise naturally when the surface is described parametrically, allowing systematic construction of the plane.For a surface expressed in parametric form, the position of any point is...

Real-World Applications of Space Curves

Real-World Applications of Space Curves

Modern aerospace navigation depends on the accurate prediction of motion in three-dimensional space. In defense applications, radar systems continuously track both interceptors and moving aerial targets to find whether their flight paths will result in a collision. These motions are modeled mathematically as space curves, which represent paths that change continuously with time. Each object’s position is described by a vector function that specifies its location in terms of time-dependent...

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by

Same journal

A Full Halin Grid Theorem.

Discrete & computational geometry·2026

Same journal

Stability and Inference of the Euler Characteristic Transform.

Discrete & computational geometry·2026

Same journal

Error Resilient Space Partitioning.

Discrete & computational geometry·2026

Same journal

A Faithful Discretization of Verbose Directional Transforms.

Discrete & computational geometry·2026

Same journal

Maximum Betti Numbers of Čech Complexes.

Discrete & computational geometry·2026

Same journal

Morse Theory for the <i>k</i>-NN Distance Function.

Discrete & computational geometry·2026

See all related articles

Search research articles

Related Experiment Video

Updated: Jun 27, 2026

Surface Mapping of Earth-like Exoplanets using Single Point Light Curves

Surface Mapping of Earth-like Exoplanets using Single Point Light Curves

Published on: May 10, 2020

Some Fast Algorithms for Curves in Surfaces.

Marc Lackenby¹

¹Mathematical Institute, University of Oxford, Woodstock Road, OX2 6GG Oxford, United Kingdom.

Discrete & Computational Geometry

|June 26, 2026

Summary

This summary is machine-generated.

New algorithms offer topological insights into curves on surfaces by efficiently computing intersection numbers and determining isotopy. These methods improve upon prior work, handling closed surfaces with polynomial time complexity relative to surface triangulation size.

More Related Videos

Characterization of Surface Modifications by White Light Interferometry: Applications in Ion Sputtering, Laser Ablation, and Tribology Experiments

Characterization of Surface Modifications by White Light Interferometry: Applications in Ion Sputtering, Laser Ablation, and Tribology Experiments

Published on: February 27, 2013

Related Experiment Videos

Last Updated: Jun 27, 2026

Surface Mapping of Earth-like Exoplanets using Single Point Light Curves

Surface Mapping of Earth-like Exoplanets using Single Point Light Curves

Published on: May 10, 2020

Characterization of Surface Modifications by White Light Interferometry: Applications in Ion Sputtering, Laser Ablation, and Tribology Experiments

Characterization of Surface Modifications by White Light Interferometry: Applications in Ion Sputtering, Laser Ablation, and Tribology Experiments

Published on: February 27, 2013

Area of Science:

Computational Topology
Differential Geometry
Surface Theory

Background:

Topological information about curves in surfaces is crucial for understanding complex geometric structures.
Existing algorithms often struggle with closed surfaces or have non-polynomial time complexities.
Efficient computation of intersection numbers and isotopy for 1-manifolds is a persistent challenge.

Purpose of the Study:

To present novel algorithms for computing topological information of curves (1-manifolds) embedded in surfaces.
To improve the efficiency and applicability of algorithms for analyzing curves in compact orientable surfaces.
To determine the geometric intersection number and isotopy of two 1-manifolds within a given surface.

Main Methods:

Development of algorithms based on surface triangulations or handle structures.
Utilizing normal form and normal coordinates for representing 1-manifolds.
Analysis of running time complexity as a polynomial function of surface size and logarithm of manifold weight.

Main Results:

An algorithm computes the geometric intersection number of two 1-manifolds in a surface with polynomial time complexity.
The algorithm efficiently handles closed surfaces, an improvement over previous methods.
A related algorithm determines if two 1-manifolds are isotopic with similar time bounds, and an algorithm for normal form conversion is provided.

Conclusions:

The presented algorithms provide significant advancements in computing topological invariants for curves on surfaces.
The improved efficiency and applicability (e.g., closed surfaces) make these methods valuable for geometric analysis.
These algorithms offer a robust framework for understanding the relationships between curves and surfaces in computational geometry.