{"version":"1.0","provider_name":"SOUL OF MATHEMATICS","provider_url":"https:\/\/soulofmathematics.com","author_name":"Rajarshi Dey","author_url":"https:\/\/soulofmathematics.com\/index.php\/author\/rajarshidey1729gmail-com\/","title":"CURVES - SOUL OF MATHEMATICS","type":"rich","width":600,"height":338,"html":"<blockquote class=\"wp-embedded-content\" data-secret=\"Nnp9Ro2FSY\"><a href=\"https:\/\/soulofmathematics.com\/index.php\/curves\/\">CURVES<\/a><\/blockquote><iframe sandbox=\"allow-scripts\" security=\"restricted\" src=\"https:\/\/soulofmathematics.com\/index.php\/curves\/embed\/#?secret=Nnp9Ro2FSY\" width=\"600\" height=\"338\" title=\"&#8220;CURVES&#8221; &#8212; SOUL OF MATHEMATICS\" data-secret=\"Nnp9Ro2FSY\" frameborder=\"0\" marginwidth=\"0\" marginheight=\"0\" scrolling=\"no\" class=\"wp-embedded-content\"><\/iframe><script type=\"text\/javascript\">\n\/* <![CDATA[ *\/\n\/*! This file is auto-generated *\/\n!function(d,l){\"use strict\";l.querySelector&&d.addEventListener&&\"undefined\"!=typeof URL&&(d.wp=d.wp||{},d.wp.receiveEmbedMessage||(d.wp.receiveEmbedMessage=function(e){var t=e.data;if((t||t.secret||t.message||t.value)&&!\/[^a-zA-Z0-9]\/.test(t.secret)){for(var s,r,n,a=l.querySelectorAll('iframe[data-secret=\"'+t.secret+'\"]'),o=l.querySelectorAll('blockquote[data-secret=\"'+t.secret+'\"]'),c=new RegExp(\"^https?:$\",\"i\"),i=0;i<o.length;i++)o[i].style.display=\"none\";for(i=0;i<a.length;i++)s=a[i],e.source===s.contentWindow&&(s.removeAttribute(\"style\"),\"height\"===t.message?(1e3<(r=parseInt(t.value,10))?r=1e3:~~r<200&&(r=200),s.height=r):\"link\"===t.message&&(r=new URL(s.getAttribute(\"src\")),n=new URL(t.value),c.test(n.protocol))&&n.host===r.host&&l.activeElement===s&&(d.top.location.href=t.value))}},d.addEventListener(\"message\",d.wp.receiveEmbedMessage,!1),l.addEventListener(\"DOMContentLoaded\",function(){for(var e,t,s=l.querySelectorAll(\"iframe.wp-embedded-content\"),r=0;r<s.length;r++)(t=(e=s[r]).getAttribute(\"data-secret\"))||(t=Math.random().toString(36).substring(2,12),e.src+=\"#?secret=\"+t,e.setAttribute(\"data-secret\",t)),e.contentWindow.postMessage({message:\"ready\",secret:t},\"*\")},!1)))}(window,document);\n\/* ]]> *\/\n<\/script>\n","thumbnail_url":"https:\/\/i1.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2020\/09\/38b5f7bfca0f997201ecbe14bacf3f08.png?fit=3635%2C2726&ssl=1","thumbnail_width":3635,"thumbnail_height":2726,"description":"Intuitively, a curve may be thought of as the trace left by a moving&nbsp;point. This is the definition that appeared more than 2000 years ago in&nbsp;Euclid&#8217;s&nbsp;Elements: &#8220;The [curved] line&nbsp;is [\u2026] 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 [\u2026] will leave from its imaginary moving some vestige in length, exempt of any width.&#8221; It is well known from elementary geometry that a line in R2 or R3 can be described by means of a parametrization t \u2192 p + tq where q not equal to 0 and p are fixed vectors, and the parameter t runs over the real numbers. Likewise, a circle in R2 (say with center 0) can be parametrized by t \u2192 (r cost, r sin t) where t \u2208 R. The common nature of these examples is expressed in the following definition. Definition. A parametrized continuous curve in Rn (n = 2, 3, . . .) is a continuous map \u03b3: I \u2192 Rn, where I \u2282 R is an open interval (of end points \u2212\u221e \u2264 a &lt; b \u2264 \u221e). The image set C = \u03b3(I) \u2282 Rn is called the trace of the curve. It is important to notice that we distinguish the curve and its trace. Physically, a curve describes the motion of a particle in n-space, and the trace is the trajectory of the particle. If the particle follows the same trajectory, but with different speed or direction, the curve is considered to be different. For example, the positive x-axis is the trace of the parametrized curve \u03b3(t) = (t, 0) where t \u2208 I =]0, \u221e[, but it is also the trace of \u02dc\u03b3(t) = (et, 0) with t \u2208 R. Example. The map \u03b3(t) = (a cost, b sin t), where a, b &gt; 0 are constants, parametrizes the ellipse Representation Of Curves The implicit and explicit way of definition. Explicit representation. Consider a Cartesian coordinate-system in E2 and the function y = f(x). Those points, the coordinates (x, f(x)) of which fulfill the equation, form a curve. This representation is called Euler-Monge-type representation of the curve. Implicit representation. Consider a Cartesian coordinate-system in E2and the function y = F(x). Those points, the coordinates (x, y) of which fulfill the equation F(x, y) = 0 form a curve. Note, that the points fulfilling the equation F(x, y) = c (where c is not equal to 0) also form a curve. This representation of the curve is introduced by Cauchy. Parametric representation. This is the method of curve representation what we described in the definition. The curve is given by the parametric form r(t) which has two (or in space three) coordinate functions: r(t) = x(t)e1 + y(t)e2. This form is frequently referred to as Gauss-type representation. The Curvature The real number is called the curvature of \u03b3 at t. Here [\u03b3\u2032(t) \u03b3\u2032\u2032(t)] denotes the 2 \u00d7 2 matrix with columns \u03b3\u2032(t) and \u03b3\u2032\u2032(t). The idea behind the definition is that the turning at t is described by the position and size of the vector \u03b3\u2032\u2032(t) relative to \u03b3\u2032(t). This relative position of the two vectors is described through their determinant, which measures the area of the parallelogram that they span. For example, if \u03b3\u2032\u2032(t) has the same direction as \u03b3\u2032(t), then the curve is not turning at all, and the determinant is zero. The power 3 in the denominator will be explained shortly by our desire to have a quantity independent of re-parametrization. MORE TO BE ADDED. STAY TUNED"}