Curves | General topology | Topology | Metric geometry
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that appeared more than 2000 years ago in Euclid's Elements: "The [curved] line is […] the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which […] will leave from its imaginary moving some vestige in length, exempt of any width." This definition of a curve has been formalized in modern mathematics as: A curve is the image of an interval to a topological space by a continuous function. In some contexts, the function that defines the curve is called a parametrization, and the curve is a parametric curve. In this article, these curves are sometimes called topological curves to distinguish them from more constrained curves such as differentiable curves. This definition encompasses most curves that are studied in mathematics; notable exceptions are level curves (which are unions of curves and isolated points), and algebraic curves (see below). Level curves and algebraic curves are sometimes called implicit curves, since they are generally defined by implicit equations. Nevertheless, the class of topological curves is very broad, and contains some curves that do not look as one may expect for a curve, or even cannot be drawn. This is the case of space-filling curves and fractal curves. For ensuring more regularity, the function that defines a curve is often supposed to be differentiable, and the curve is then said to be a differentiable curve. A plane algebraic curve is the zero set of a polynomial in two indeterminates. More generally, an algebraic curve is the zero set of a finite set of polynomials, which satisfies the further condition of being an algebraic variety of dimension one. If the coefficients of the polynomials belong to a field k, the curve is said to be defined over k. In the common case of a real algebraic curve, where k is the field of real numbers, an algebraic curve is a finite union of topological curves. When complex zeros are considered, one has a complex algebraic curve, which, from the topological point of view, is not a curve, but a surface, and is often called a Riemann surface. Although not being curves in the common sense, algebraic curves defined over other fields have been widely studied. In particular, algebraic curves over a finite field are widely used in modern cryptography. (Wikipedia).
#Cycloid: A curve traced by a point on a circle rolling in a straight line. (A preview of this Sunday's video.)
From playlist Miscellaneous
11_6_1 Contours and Tangents to Contours Part 1
A contour is simply the intersection of the curve of a function and a plane or hyperplane at a specific level. The gradient of the original function is a vector perpendicular to the tangent of the contour at a point on the contour.
From playlist Advanced Calculus / Multivariable Calculus
Find the point where their exist a horizontal tangent line
👉 Learn how to find the point of the horizontal tangent of a curve. A tangent to a curve is a line that touches a point in the outline of the curve. When given a curve described by the function y = f(x). The value of x for which the derivative of the function y, is zero is the point of hor
From playlist Find the Point Where the Tangent Line is Horizontal
How do you graph an equation using the intercept method
👉 Learn about graphing linear equations. A linear equation is an equation whose highest exponent on its variable(s) is 1. i.e. linear equations has no exponents on their variables. The graph of a linear equation is a straight line. To graph a linear equation, we identify two values (x-valu
From playlist ⚡️Graph Linear Equations | Learn About
Free ebook http://tinyurl.com/EngMathYT How to integrate over 2 curves. This example discusses the additivity property of line integrals (sometimes called path integrals).
From playlist Engineering Mathematics
Calculus - What is a Derivative? (3 of 8) Slope of a Tangent Line to a Curve
Visit http://ilectureonline.com for more math and science lectures! In this video I will explain the slope of a tangent line to a curve.
From playlist CALCULUS 1 CH 2 WHAT IS A DERIVATIVE?
How to calculate arc length of a curve.
Free ebook http://tinyurl.com/EngMathYT How to calculate the arc length of a curve: a basic example.
From playlist A second course in university calculus.
Find the values where the function has horizontal tangents
👉 Learn how to find the point of the horizontal tangent of a curve. A tangent to a curve is a line that touches a point in the outline of the curve. When given a curve described by the function y = f(x). The value of x for which the derivative of the function y, is zero is the point of hor
From playlist Find the Point Where the Tangent Line is Horizontal
Maria Montanucci: Algebraic curves with many rational points over finite fields
CONFERENCE Recording during the thematic meeting : « Conference On alGebraic varieties over fiNite fields and Algebraic geometry Codes» the February 13, 2023 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Jean Petit Find this video and other talks
From playlist Algebraic and Complex Geometry
What is the slope of a linear equation
👉 Learn about graphing linear equations. A linear equation is an equation whose highest exponent on its variable(s) is 1. i.e. linear equations has no exponents on their variables. The graph of a linear equation is a straight line. To graph a linear equation, we identify two values (x-valu
From playlist ⚡️Graph Linear Equations | Learn About
Lecture 10: Smooth Curves (Discrete Differential Geometry)
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS For more information see http://geometry.cs.cmu.edu/ddg
From playlist Discrete Differential Geometry - CMU 15-458/858
Binbin Xu: Equivalent curves on surfaces
We consider a closed oriented surface of genus at least 2. For any positive integer k, an essential closed curve on the surface with k self-intersections is called a k-curve. A pair of curves on the surface are said to be k-equivalent, if they have the same intersection numbers with each k
From playlist Topology
Lecture 11: Discrete Curves (Discrete Differential Geometry)
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS For more information see http://geometry.cs.cmu.edu/ddg
From playlist Discrete Differential Geometry - CMU 15-458/858
Ö. Yurttas - Algorithms for multicurves with Dynnikov coordinates
Multicurves have played a fundamental role in the study of mapping class groups of surfaces since the work of Dehn. A beautiful method of describing such systems on the n-punctured disk is given by the Dynnikov coordinate system. In this talk we describe polynomial time algorithms for cal
From playlist Ecole d'été 2018 - Teichmüller dynamics, mapping class groups and applications
CTNT 2020 - 3-adic images of Galois for elliptic curves over Q - Jeremy Rouse
The Connecticut Summer School in Number Theory (CTNT) is a summer school in number theory for advanced undergraduate and beginning graduate students, to be followed by a research conference. For more information and resources please visit: https://ctnt-summer.math.uconn.edu/
From playlist CTNT 2020 - Conference Videos
What is a Tensor? Lesson 21: The Lie derivative
What is a Tensor? Lesson 21: The Lie derivative We reconstruct the notion of a vector space at a point in spacetime using the more fundamental exposition of tangent vectors to curves. Then we define a congruence of curves associated with a vector field and then we define the Lie derivativ
From playlist What is a Tensor?
Hierarchy Hyperbolic Spaces (Lecture – 01) by Jason Behrstock
Geometry, Groups and Dynamics (GGD) - 2017 DATE: 06 November 2017 to 24 November 2017 VENUE: Ramanujan Lecture Hall, ICTS, Bengaluru The program focuses on geometry, dynamical systems and group actions. Topics are chosen to cover the modern aspects of these areas in which research has b
From playlist Geometry, Groups and Dynamics (GGD) - 2017
Abbey Bourdon : Minimal torsion curves in geometric isogeny classes
CONFERENCE Recording during the thematic meeting : "COUNT, COmputations and their Uses in Number Theory" the March 02, 2023 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide math
From playlist JEAN MORLET CHAIR
What are the x and y intercepts of a linear equation
👉 Learn about graphing linear equations. A linear equation is an equation whose highest exponent on its variable(s) is 1. i.e. linear equations has no exponents on their variables. The graph of a linear equation is a straight line. To graph a linear equation, we identify two values (x-valu
From playlist ⚡️Graph Linear Equations | Learn About
"From Diophantus to Bitcoin: Why Are Elliptic Curves Everywhere?" by Alvaro Lozano-Robledo
This talk was organized by the Number Theory Unit of the Center for Advanced Mathematical Sciences at the American University of Beirut, on November 1st, 2022. Abstract: Elliptic curves are ubiquitous in number theory, algebraic geometry, complex analysis, cryptography, physics, and beyo
From playlist Math Talks