Calculus III: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=158
en-us2015 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemFri, 30 Jan 2015 07:56:35 GMTFri, 30 Jan 2015 07:56:35 GMTNew applications in the Calculus III categoryhttp://www.mapleprimes.com/images/mapleapps.gifCalculus III: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=158
The Comet 67P/Churyumov-Gerasimenko, Rosetta & Philae
http://www.maplesoft.com/applications/view.aspx?SID=153706&ref=Feed
<p> Abstract<br /><br />The Rosetta space probe launched 10 years ago by the European Space Agency (ESA) arrived recently (November 12, 2014) at the site of the comet known as 67P/Churyumov-Gerasimenco after a trip of 4 billions miles from Earth. After circling the comet, Rosetta released its precious load : the lander Philae packed with 21 different scientific instruments for the study of the comet with the main purpose : the origin of our solar system and possibly the origin of life on our planet.<br /><br />Our plan is rather a modest one since all we want is to get , by calculations, specific data concerning the comet and its lander.<br />We shall take a simplified model and consider the comet as a perfect solid sphere to which we can apply Newton's laws.<br /><br />We want to find:<br /><br />I- the acceleration on the comet surface ,<br />II- its radius,<br />III- its density,<br />IV- the velocity of Philae just after the 1st bounce off the comet (it has bounced twice),<br />V- the time for Philae to reach altitude of 1000 m above the comet .<br /><br />We shall compare our findings with the already known data to see how close our simplified mathematical model findings are to the duck-shaped comet already known results.<br />It turned out that our calculations for a sphere shaped comet are very close to the already known data.<br /><br />Conclusion<br /><br />Even with a shape that defies the application of any mechanical laws we can always get very close to reality by adopting a simplified mathematical model in any preliminary study of a complicated problem.<br /><br /></p><img src="/applications/images/app_image_blank_lg.jpg" alt="The Comet 67P/Churyumov-Gerasimenko, Rosetta & Philae" align="left"/><p> Abstract<br /><br />The Rosetta space probe launched 10 years ago by the European Space Agency (ESA) arrived recently (November 12, 2014) at the site of the comet known as 67P/Churyumov-Gerasimenco after a trip of 4 billions miles from Earth. After circling the comet, Rosetta released its precious load : the lander Philae packed with 21 different scientific instruments for the study of the comet with the main purpose : the origin of our solar system and possibly the origin of life on our planet.<br /><br />Our plan is rather a modest one since all we want is to get , by calculations, specific data concerning the comet and its lander.<br />We shall take a simplified model and consider the comet as a perfect solid sphere to which we can apply Newton's laws.<br /><br />We want to find:<br /><br />I- the acceleration on the comet surface ,<br />II- its radius,<br />III- its density,<br />IV- the velocity of Philae just after the 1st bounce off the comet (it has bounced twice),<br />V- the time for Philae to reach altitude of 1000 m above the comet .<br /><br />We shall compare our findings with the already known data to see how close our simplified mathematical model findings are to the duck-shaped comet already known results.<br />It turned out that our calculations for a sphere shaped comet are very close to the already known data.<br /><br />Conclusion<br /><br />Even with a shape that defies the application of any mechanical laws we can always get very close to reality by adopting a simplified mathematical model in any preliminary study of a complicated problem.<br /><br /></p>153706Mon, 17 Nov 2014 05:00:00 ZDr. Ahmed BaroudyDr. Ahmed BaroudyGuia de estudio para integrales dobles
http://www.maplesoft.com/applications/view.aspx?SID=153595&ref=Feed
<p>Esta guía de estudio tiene como objetivo aprovechar las capacidades de Maple para generar gráficas interactivas y lograr con ellas que el estudiante comprenda el problema geométrico que da origen a la integral doble, la interpretación geométrica de una integral doble cuando el integrando es positivo, y la interpretación geométrica del cálculo de integrales iteradas en una integral doble.</p><img src="/view.aspx?si=153595/Preview_figure.png" alt="Guia de estudio para integrales dobles" align="left"/><p>Esta guía de estudio tiene como objetivo aprovechar las capacidades de Maple para generar gráficas interactivas y lograr con ellas que el estudiante comprenda el problema geométrico que da origen a la integral doble, la interpretación geométrica de una integral doble cuando el integrando es positivo, y la interpretación geométrica del cálculo de integrales iteradas en una integral doble.</p>153595Tue, 03 Jun 2014 04:00:00 ZDr. Ranferi GutierrezDr. Ranferi GutierrezGuia electronica de estudio sobre Multiplicadores de Lagrange
http://www.maplesoft.com/applications/view.aspx?SID=153587&ref=Feed
<p>El objetivo principal de la presente guia electronica de estudio es que el estudiante adquiera una comprension geometrica de lo que significa buscar valores extremos de una funcion de varias variables, la cual esta sujeta a una restriccion. El objetivo principal de esta guia se logra aprovechando las capacidades de Maple para generar graficos en 3D y manipularlos a traves del uso de componentes.</p><img src="/view.aspx?si=153587/multiplicadores.png" alt="Guia electronica de estudio sobre Multiplicadores de Lagrange" align="left"/><p>El objetivo principal de la presente guia electronica de estudio es que el estudiante adquiera una comprension geometrica de lo que significa buscar valores extremos de una funcion de varias variables, la cual esta sujeta a una restriccion. El objetivo principal de esta guia se logra aprovechando las capacidades de Maple para generar graficos en 3D y manipularlos a traves del uso de componentes.</p>153587Tue, 20 May 2014 04:00:00 ZDr. Ranferi GutierrezDr. Ranferi GutierrezClassroom Tips and Techniques: Drawing a Normal and Tangent Plane on a Surface
http://www.maplesoft.com/applications/view.aspx?SID=150722&ref=Feed
Four different techniques are given for obtaining a graph showing a surface with a normal and tangent plane attached. The work is a response to <a href="http://www.mapleprimes.com/questions/147681-A-Problem-About-Plot-The-Part-Of-The-Surface">a MaplePrimes question asked on May 25, 2013</a>.<img src="/view.aspx?si=150722/thumb.jpg" alt="Classroom Tips and Techniques: Drawing a Normal and Tangent Plane on a Surface" align="left"/>Four different techniques are given for obtaining a graph showing a surface with a normal and tangent plane attached. The work is a response to <a href="http://www.mapleprimes.com/questions/147681-A-Problem-About-Plot-The-Part-Of-The-Surface">a MaplePrimes question asked on May 25, 2013</a>.150722Tue, 20 Aug 2013 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: New Tools for Lines and Planes
http://www.maplesoft.com/applications/view.aspx?SID=144642&ref=Feed
The fifteen new "Lines and Planes" commands in the Student MultivariateCalculus package are detailed, and then illustrated via a collection of examples from a typical calculus course. These new commands can also be implemented through the Context Menu system, as shown by parallel solutions in the set of examples.<img src="/view.aspx?si=144642/thumb.jpg" alt="Classroom Tips and Techniques: New Tools for Lines and Planes" align="left"/>The fifteen new "Lines and Planes" commands in the Student MultivariateCalculus package are detailed, and then illustrated via a collection of examples from a typical calculus course. These new commands can also be implemented through the Context Menu system, as shown by parallel solutions in the set of examples.144642Thu, 14 Mar 2013 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Animated Trace of a Curve Drawn by Radius Vector
http://www.maplesoft.com/applications/view.aspx?SID=143371&ref=Feed
A plane curve <strong>R</strong>(<em>t</em>) = <em>x</em>(<em>t</em>) <strong>i</strong> + <em>y</em>(<em>t</em>) <strong>j</strong> is traced by a "moving" radius vector <strong>R</strong>(<em>t</em>). Code for this animation is explored in this article.<img src="/view.aspx?si=143371/thumb.jpg" alt="Classroom Tips and Techniques: Animated Trace of a Curve Drawn by Radius Vector" align="left"/>A plane curve <strong>R</strong>(<em>t</em>) = <em>x</em>(<em>t</em>) <strong>i</strong> + <em>y</em>(<em>t</em>) <strong>j</strong> is traced by a "moving" radius vector <strong>R</strong>(<em>t</em>). Code for this animation is explored in this article.143371Mon, 11 Feb 2013 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Least-Squares Fits
http://www.maplesoft.com/applications/view.aspx?SID=140942&ref=Feed
<p><span id="ctl00_mainContent__documentViewer" ><span ><span class="body summary">The least-squares fitting of functions to data can be done in Maple with eleven different commands from four different packages. The <em>CurveFitting</em> and LinearAlgebra packages each have a LeastSquares command; the Optimization package has the LSSolve and NLPSolve commands; and the Statistics package has the seven commands Fit, LinearFit, PolynomialFit, ExponentialFit, LogarithmicFit, PowerFit, and NonlinearFit, which can return some measure of regression analysis.</span></span></span></p><img src="/view.aspx?si=140942/image.jpg" alt="Classroom Tips and Techniques: Least-Squares Fits" align="left"/><p><span id="ctl00_mainContent__documentViewer" ><span ><span class="body summary">The least-squares fitting of functions to data can be done in Maple with eleven different commands from four different packages. The <em>CurveFitting</em> and LinearAlgebra packages each have a LeastSquares command; the Optimization package has the LSSolve and NLPSolve commands; and the Statistics package has the seven commands Fit, LinearFit, PolynomialFit, ExponentialFit, LogarithmicFit, PowerFit, and NonlinearFit, which can return some measure of regression analysis.</span></span></span></p>140942Wed, 28 Nov 2012 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: An Inequality-Constrained Optimization Problem
http://www.maplesoft.com/applications/view.aspx?SID=135904&ref=Feed
<p>This article shows how to work both analytically and numerically to find the global maximum of</p>
<p><em>w</em> = ƒ(<em>x, y, z</em>) ≡ <em>x</em><sup>2</sup>(1 + <em>x</em>) + <em>y</em><sup>2</sup>(1 + <em>y</em>) + z<sup>2</sup>(1 + <em>z</em>)</p>
<p>in that part of the first octant on, or below, the plane <em>x</em> + <em>y</em> + <em>z</em> = 6.</p><img src="/view.aspx?si=135904/thumb.jpg" alt="Classroom Tips and Techniques: An Inequality-Constrained Optimization Problem" align="left"/><p>This article shows how to work both analytically and numerically to find the global maximum of</p>
<p><em>w</em> = ƒ(<em>x, y, z</em>) ≡ <em>x</em><sup>2</sup>(1 + <em>x</em>) + <em>y</em><sup>2</sup>(1 + <em>y</em>) + z<sup>2</sup>(1 + <em>z</em>)</p>
<p>in that part of the first octant on, or below, the plane <em>x</em> + <em>y</em> + <em>z</em> = 6.</p>135904Mon, 16 Jul 2012 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Custom and Task Palettes
http://www.maplesoft.com/applications/view.aspx?SID=132914&ref=Feed
New in Maple 16, the Custom palette is a palette added to Maple by the user. It is populated with task templates that are already in the Task Browser Table of Contents. A separate Tasks palette can be populated with task templates created by the "Create Task" option in the Context Menu for any selection in a worksheet. This article sheds light on these new functionalities, and gives an example of a Custom palette developed to capture part of the geom3d package in task templates.<img src="/view.aspx?si=132914/thumb.jpg" alt="Classroom Tips and Techniques: Custom and Task Palettes" align="left"/>New in Maple 16, the Custom palette is a palette added to Maple by the user. It is populated with task templates that are already in the Task Browser Table of Contents. A separate Tasks palette can be populated with task templates created by the "Create Task" option in the Context Menu for any selection in a worksheet. This article sheds light on these new functionalities, and gives an example of a Custom palette developed to capture part of the geom3d package in task templates.132914Thu, 12 Apr 2012 04:00:00 ZDr. Robert LopezDr. Robert LopezSpherical Pendulum with Animation
http://www.maplesoft.com/applications/view.aspx?SID=132143&ref=Feed
<p>Some years ago I have written a Maple document ( already on Maple's online) on the subject of animating a simple pendulum for large angles of oscillation. This gave me the chance to test Maple command JacobiSN(time, k). I was very much pleased to see Maple do a wonderful job in getting these Jacobi's elliptic functions without a glitch.<br />Today I am back to these same functions for a similar purpose though much more sophisticated than the previous one.<br />The idea is:<br />1- to get the differential equations of motion for the Spherical Pendulum (SP),<br />2- to solve them,<br />3- to use Maple for finding the inverse of these Elliptic Integrals i.e. finding the displacement z as function of time,<br />4- to get a set of coordinates [x, y, z] for the positions of the bob at different times for plotting,<br />5- finally to work out the necessary steps for the purpose of animation.<br />It turns out that even with only 3 oscillations where each is defined with only 20 positions of the bob for a total of 60 points on the graph, the animation is so overwhelming that Maple reports:<br /> " the length of the output exceeds 1 million".<br />Not withstanding this warning, Maple did a perfect job by getting the animation to my satisfaction. <br />Note that with only 60 positions of the bob, the present article length is equal to 11.3 MB! To be able to upload it, I have to save it without running the last command related to the animation. Doing so I reduced it to a mere 570 KB.<br /><br />It was tiring to get through a jumble of formulas, calculations and programming so I wonder why I have to go through all this trouble to get this animation and yet one can get the same thing with much better animation from the internet. I think the reason is the challenge to be able to do things that others have done before and secondly the idea of creating something form nothing then to see it working as expected, gives (at least to me) a great deal of pleasure and satisfaction.<br />This is beside the fact that, to my knowledge, no such animation for (SP) has been published on Maple online with detailed calculations & programming as I did.<br /><br /></p><img src="/view.aspx?si=132143/433082\Spherical_Pendulum_p.jpg" alt="Spherical Pendulum with Animation" align="left"/><p>Some years ago I have written a Maple document ( already on Maple's online) on the subject of animating a simple pendulum for large angles of oscillation. This gave me the chance to test Maple command JacobiSN(time, k). I was very much pleased to see Maple do a wonderful job in getting these Jacobi's elliptic functions without a glitch.<br />Today I am back to these same functions for a similar purpose though much more sophisticated than the previous one.<br />The idea is:<br />1- to get the differential equations of motion for the Spherical Pendulum (SP),<br />2- to solve them,<br />3- to use Maple for finding the inverse of these Elliptic Integrals i.e. finding the displacement z as function of time,<br />4- to get a set of coordinates [x, y, z] for the positions of the bob at different times for plotting,<br />5- finally to work out the necessary steps for the purpose of animation.<br />It turns out that even with only 3 oscillations where each is defined with only 20 positions of the bob for a total of 60 points on the graph, the animation is so overwhelming that Maple reports:<br /> " the length of the output exceeds 1 million".<br />Not withstanding this warning, Maple did a perfect job by getting the animation to my satisfaction. <br />Note that with only 60 positions of the bob, the present article length is equal to 11.3 MB! To be able to upload it, I have to save it without running the last command related to the animation. Doing so I reduced it to a mere 570 KB.<br /><br />It was tiring to get through a jumble of formulas, calculations and programming so I wonder why I have to go through all this trouble to get this animation and yet one can get the same thing with much better animation from the internet. I think the reason is the challenge to be able to do things that others have done before and secondly the idea of creating something form nothing then to see it working as expected, gives (at least to me) a great deal of pleasure and satisfaction.<br />This is beside the fact that, to my knowledge, no such animation for (SP) has been published on Maple online with detailed calculations & programming as I did.<br /><br /></p>132143Mon, 26 Mar 2012 04:00:00 ZDr. Ahmed BaroudyDr. Ahmed BaroudyParameterizing Motion along a Curve
http://www.maplesoft.com/applications/view.aspx?SID=130465&ref=Feed
<p>We use the Euler-Lagrange equation to parameterize the motion of a bead on a parabola, a helix, and a piecewise defined combination of the two.</p><img src="/applications/images/app_image_blank_lg.jpg" alt="Parameterizing Motion along a Curve" align="left"/><p>We use the Euler-Lagrange equation to parameterize the motion of a bead on a parabola, a helix, and a piecewise defined combination of the two.</p>130465Wed, 08 Feb 2012 05:00:00 ZShawn HedmanShawn HedmanClassroom Tips and Techniques: Directional Derivatives in Maple
http://www.maplesoft.com/applications/view.aspx?SID=126623&ref=Feed
Several identities in vector calculus involve the operator A . (VectorCalculus[Nabla]) acting on a vector B. The resulting expression (A . (VectorCalculus[Nabla]))B is interpreted as the directional derivative of the vector B in the direction of the vector A. This is not easy to implement in Maple's VectorCalculus packages. However, this functionality exists in the Physics:-Vectors package, and in the DifferentialGeometry package where it is properly called the DirectionalCovariantDerivative.
This article examines how to obtain (A . (VectorCalculus[Nabla]))B in Maple.<img src="/view.aspx?si=126623/thumb.jpg" alt="Classroom Tips and Techniques: Directional Derivatives in Maple" align="left"/>Several identities in vector calculus involve the operator A . (VectorCalculus[Nabla]) acting on a vector B. The resulting expression (A . (VectorCalculus[Nabla]))B is interpreted as the directional derivative of the vector B in the direction of the vector A. This is not easy to implement in Maple's VectorCalculus packages. However, this functionality exists in the Physics:-Vectors package, and in the DifferentialGeometry package where it is properly called the DirectionalCovariantDerivative.
This article examines how to obtain (A . (VectorCalculus[Nabla]))B in Maple.126623Fri, 14 Oct 2011 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Gems 16-20 from the Red Book of Maple Magic
http://www.maplesoft.com/applications/view.aspx?SID=125886&ref=Feed
From the Red Book of Maple Magic, Gems 16-20: Vectors with assumptions in VectorCalculus, aliasing commands to symbols, setting iterated integrals from the Expression palette, writing a slider value to a label, and writing text to a math container.<img src="/view.aspx?si=125886/thumb.jpg" alt="Classroom Tips and Techniques: Gems 16-20 from the Red Book of Maple Magic" align="left"/>From the Red Book of Maple Magic, Gems 16-20: Vectors with assumptions in VectorCalculus, aliasing commands to symbols, setting iterated integrals from the Expression palette, writing a slider value to a label, and writing text to a math container.125886Fri, 23 Sep 2011 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Steepest-Ascent Curves
http://www.maplesoft.com/applications/view.aspx?SID=123985&ref=Feed
Steepest-ascent curves are obtained for surfaces defined analytically and digitally.<img src="/view.aspx?si=123985/thumb.jpg" alt="Classroom Tips and Techniques: Steepest-Ascent Curves" align="left"/>Steepest-ascent curves are obtained for surfaces defined analytically and digitally.123985Tue, 19 Jul 2011 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Nonlinear Fit, Optimization, and the DirectSearch Package
http://www.maplesoft.com/applications/view.aspx?SID=122760&ref=Feed
In this month's article, I revisit a nonlinear curve-fitting problem that appears in my Advanced Engineering Mathematics ebook, examine the role of Maple's Optimization package in that problem, and then explore the DirectSearch package from Dr. Sergey N. Moiseev.<img src="/view.aspx?si=122760/thumb.jpg" alt="Classroom Tips and Techniques: Nonlinear Fit, Optimization, and the DirectSearch Package" align="left"/>In this month's article, I revisit a nonlinear curve-fitting problem that appears in my Advanced Engineering Mathematics ebook, examine the role of Maple's Optimization package in that problem, and then explore the DirectSearch package from Dr. Sergey N. Moiseev.122760Wed, 15 Jun 2011 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: More Gems from the Little Red Book of Maple Magic
http://www.maplesoft.com/applications/view.aspx?SID=101922&ref=Feed
Five more bits of "Maple magic" accumulated in recent months are shared: "if" with certain exact numbers, constant functions, replacing a product with a name, assumptions on subscripted variables, and gradient vectors via the Matrix palette.<img src="/view.aspx?si=101922/thumb.jpg" alt="Classroom Tips and Techniques: More Gems from the Little Red Book of Maple Magic" align="left"/>Five more bits of "Maple magic" accumulated in recent months are shared: "if" with certain exact numbers, constant functions, replacing a product with a name, assumptions on subscripted variables, and gradient vectors via the Matrix palette.101922Tue, 22 Feb 2011 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Gems from the Little Red Book of Maple Magic
http://www.maplesoft.com/applications/view.aspx?SID=100897&ref=Feed
Five bits of "Maple magic" accumulated in recent months are shared: converting the half-angle trig formulas to radicals, tickmarks along a parametric curve, writing unevaluated math on a graph, changing Maple's differentiation formulas, and drawing a decent surface for a function containing a square root.<img src="/view.aspx?si=100897/thumb.jpg" alt="Classroom Tips and Techniques: Gems from the Little Red Book of Maple Magic" align="left"/>Five bits of "Maple magic" accumulated in recent months are shared: converting the half-angle trig formulas to radicals, tickmarks along a parametric curve, writing unevaluated math on a graph, changing Maple's differentiation formulas, and drawing a decent surface for a function containing a square root.100897Fri, 14 Jan 2011 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Partial Derivatives by Subscripting
http://www.maplesoft.com/applications/view.aspx?SID=100266&ref=Feed
As output, Maple can display the partial derivative ∂/∂<em>x f</em>(<em>x,y</em>) as <em>f</em><sub>x</sub>; that is, subscript notation can be used to display partial derivatives, and it can be done with two completely different mechanisms. This article describes these two techniques, and then investigates the extent to which partial derivatives can be calculated by subscript notation.<img src="/view.aspx?si=100266/thumb.jpg" alt="Classroom Tips and Techniques: Partial Derivatives by Subscripting" align="left"/>As output, Maple can display the partial derivative ∂/∂<em>x f</em>(<em>x,y</em>) as <em>f</em><sub>x</sub>; that is, subscript notation can be used to display partial derivatives, and it can be done with two completely different mechanisms. This article describes these two techniques, and then investigates the extent to which partial derivatives can be calculated by subscript notation.100266Wed, 15 Dec 2010 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Visualizing Regions of Integration
http://www.maplesoft.com/applications/view.aspx?SID=94845&ref=Feed
<p>Five of the new task templates in Maple 14 are designed to help visualize regions of integration for iterated integrals. In particular, there are task templates for double integrals in Cartesian and polar coordinates, and for triple integrals in Cartesian, cylindrical, and spherical coordinates. These task templates can be found at the end of the path</p>
<p>Tools ≻ Tasks ≻ Browse: Calculus - Multivariate ≻ Integration ≻ Visualizing Regions of Integration</p>
<p>Each of these task templates provides for iterating the relevant multiple integral in any of its possible orders. An example for each task template is provided.</p><img src="/view.aspx?si=94845/thumb.jpg" alt="Classroom Tips and Techniques: Visualizing Regions of Integration" align="left"/><p>Five of the new task templates in Maple 14 are designed to help visualize regions of integration for iterated integrals. In particular, there are task templates for double integrals in Cartesian and polar coordinates, and for triple integrals in Cartesian, cylindrical, and spherical coordinates. These task templates can be found at the end of the path</p>
<p>Tools ≻ Tasks ≻ Browse: Calculus - Multivariate ≻ Integration ≻ Visualizing Regions of Integration</p>
<p>Each of these task templates provides for iterating the relevant multiple integral in any of its possible orders. An example for each task template is provided.</p>94845Tue, 06 Jul 2010 04:00:00 ZDr. Robert LopezDr. Robert LopezThe Astroid and Its Tangent Lines
http://www.maplesoft.com/applications/view.aspx?SID=89485&ref=Feed
<p>Tangents to the astroid have fixed length if restricted to a single quadrant. In fact, the envelope of a line of fixed length that has its endpoints on the coordinate axes in the first quadrant is just the astroid. In this month's article, we explore these two aspects of the astroid, a curve also known as the four-cusped hypocycloid, as well as the tetracuspid, cubocycloid, and paracycle.</p><img src="/view.aspx?si=89485/thumb.jpg" alt="The Astroid and Its Tangent Lines" align="left"/><p>Tangents to the astroid have fixed length if restricted to a single quadrant. In fact, the envelope of a line of fixed length that has its endpoints on the coordinate axes in the first quadrant is just the astroid. In this month's article, we explore these two aspects of the astroid, a curve also known as the four-cusped hypocycloid, as well as the tetracuspid, cubocycloid, and paracycle.</p>89485Thu, 10 Jun 2010 04:00:00 ZDr. Robert LopezDr. Robert Lopez