Problem 42580. Conic equation

Created by Tim

Solve Later

{"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","targetMode":"","relationshipId":"rId1","target":"/matlab/document.xml"},{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/output","targetMode":"","relationshipId":"rId2","target":"/matlab/output.xml"}],"parts":[{"partUri":"/matlab/document.xml","relationship":[],"contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n<w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>A conic of revolution (around the</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>z</w:t></w:r><w:r><w:t> axis) can be defined by the equation</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ s^2 – 2*R*z + (k+1)*z^2 = 0]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>where</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>s^2=x^2+y^2</w:t></w:r><w:r><w:t>,</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>R</w:t></w:r><w:r><w:t> is the vertex radius of curvature, and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>k</w:t></w:r><w:r><w:t> is the conic constant:</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>k&lt;-1</w:t></w:r><w:r><w:t> for a hyperbola,</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>k=-1</w:t></w:r><w:r><w:t> for a parabola,</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>-1&lt;k&lt;0</w:t></w:r><w:r><w:t> for a tall ellipse,</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>k=0</w:t></w:r><w:r><w:t> for a sphere, and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>k&gt;0</w:t></w:r><w:r><w:t> for a short ellipse.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Write a function</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>z=conic(s,R,k)</w:t></w:r><w:r><w:t> to calculate height</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>z</w:t></w:r><w:r><w:t> as a function of radius</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>s</w:t></w:r><w:r><w:t> for given</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>R</w:t></w:r><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>k</w:t></w:r><w:r><w:t>. Choose the branch of the solution that gives</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>z=s^2/(2*R)+...</w:t></w:r><w:r><w:t> for small values of</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>s</w:t></w:r><w:r><w:t>. This defines a concave surface for</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>R&gt;0</w:t></w:r><w:r><w:t> and a convex surface for</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>R&lt;0</w:t></w:r><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>The trick is to get full machine precision for all values of</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>s</w:t></w:r><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>R</w:t></w:r><w:r><w:t>. The test suite will require a relative error less than</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>4*eps</w:t></w:r><w:r><w:t>, where</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>eps</w:t></w:r><w:r><w:t> is the machine precision.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Hint (added 2015/09/03): the straightforward solution is</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ z = (R-sqrt(R^2-(k+1)*s^2))/(k+1),]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>but this does not work if</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>k=-1</w:t></w:r><w:r><w:t>, gives the wrong branch of the solution if</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>R&lt;0</w:t></w:r><w:r><w:t>, and is subject to severe roundoff error if</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>s^2</w:t></w:r><w:r><w:t> is small compared to</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>R^2</w:t></w:r><w:r><w:t>. It is possible, however, to find a mathematically equivalent form of the solution that solves all three problems at once.</w:t></w:r></w:p></w:body></w:document>"},{"partUri":"/matlab/output.xml","contentType":"text/xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\" ?><embeddedOutputs><metaData><evaluationState>manual</evaluationState><layoutState>code</layoutState><outputStatus>ready</outputStatus></metaData><outputArray type=\"array\"/><regionArray type=\"array\"/></embeddedOutputs>"}]}

A conic of revolution (around the <tt>z</tt> axis) can be defined by the equation<pre> s^2 – 2*R*z + (k+1)*z^2 = 0</pre>where <tt>s^2=x^2+y^2</tt>, <tt>R</tt> is the vertex radius of curvature, and <tt>k</tt> is the conic constant: <tt>k&lt;-1</tt> for a hyperbola, <tt>k=-1</tt> for a parabola, <tt>-1&lt;k&lt;0</tt> for a tall ellipse, <tt>k=0</tt> for a sphere, and <tt>k&gt;0</tt> for a short ellipse.Write a function <tt>z=conic(s,R,k)</tt> to calculate height <tt>z</tt> as a function of radius <tt>s</tt> for given <tt>R</tt> and <tt>k</tt>. Choose the branch of the solution that gives <tt>z=s^2/(2*R)+...</tt> for small values of <tt>s</tt>. This defines a concave surface for <tt>R&gt;0</tt> and a convex surface for <tt>R&lt;0</tt>.The trick is to get full machine precision for all values of <tt>s</tt> and <tt>R</tt>. The test suite will require a relative error less than <tt>4*eps</tt>, where <tt>eps</tt> is the machine precision.Hint (added 2015/09/03): the straightforward solution is<pre> z = (R-sqrt(R^2-(k+1)*s^2))/(k+1), </pre>but this does not work if <tt>k=-1</tt>, gives the wrong branch of the solution if <tt>R&lt;0</tt>, and is subject to severe roundoff error if <tt>s^2</tt> is small compared to <tt>R^2</tt>. It is possible, however, to find a mathematically equivalent form of the solution that solves all three problems at once.

Solve

Solution Stats

52 Solutions
15 Solvers

Last Solution submitted on Dec 01, 2020

Last 200 Solutions

Problem Comments

1 Comment

1 Comment

Rafael S.T. Vieira on 30 Jun 2020

good one.

Problem Recent Solvers15

Problem Tags

conic quadratic

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!

Problem 42580. Conic equation

Solution Stats

Last 200 Solutions

Problem Comments

Problem Recent Solvers15

Suggested Problems

More from this Author10

Problem Tags

Community Treasure Hunt

Problem 42580. Conic equation

Solution Stats

Last 200 Solutions

Problem Comments

Problem Recent Solvers15

Suggested Problems

More from this Author10

Problem Tags

Community Treasure Hunt

Players