TangentPlane - Maple Help

RegularChains[AlgebraicGeometryTools]

 TangentPlane
 compute the plane plane of an hypersurface at a point given by a regular chain

 Calling Sequence TangentPlane(rc, f, R)

Parameters

 R - polynomial ring rc - regular chain of R f - a polynomial of R

Description

 • The command TangentPlane(rc, f, R) returns the tangent plane of the hypersurface defined by f at every point of F defined by rc.
 • The result is a list of pairs [g,ts] where ts is a zero-dimensional regular chain the zero set of which is contained in that g, and g a polynomial the zero set of which defines the tangent plane of f at ts.
 • It is assumed that rc is a zero-dimensional regular chain.
 • It is assumed that the hypersurface defined by f is non-singular at every point defined by rc.
 • This command is part of the RegularChains[AlgebraicGeometryTools] package, so it can be used in the form TangentPlane(..) only after executing the command with(RegularChains[AlgebraicGeometryTools]).  However, it can always be accessed through the long form of the command by using RegularChains[AlgebraicGeometryTools][TangentPlane](..).

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$$\mathrm{with}\left(\mathrm{ChainTools}\right):$$\mathrm{with}\left(\mathrm{AlgebraicGeometryTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,z\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)
 > $\mathrm{rc}≔\mathrm{Empty}\left(R\right)$
 ${\mathrm{rc}}{≔}{\mathrm{regular_chain}}$ (2)
 > $\mathrm{rc}≔\mathrm{Chain}\left(\left[z-1,y,x\right],\mathrm{rc},R\right)$
 ${\mathrm{rc}}{≔}{\mathrm{regular_chain}}$ (3)
 > $\mathrm{Equations}\left(\mathrm{rc},R\right)$
 $\left[{x}{,}{y}{,}{z}{-}{1}\right]$ (4)
 > $f≔xz+zy+yx$
 ${f}{≔}{y}{}{x}{+}{x}{}{z}{+}{z}{}{y}$ (5)
 > $\mathrm{tp}≔\mathrm{TangentPlane}\left(\mathrm{rc},f,R\right)$
 ${\mathrm{tp}}{≔}\left[\left[{\mathrm{_x}}{+}{\mathrm{_y}}{,}{\mathrm{rc}}\right]\right]$ (6)

Compatibility

 • The RegularChains[AlgebraicGeometryTools][TangentPlane] command was introduced in Maple 2020.