Plane

Author

Eric W. Weisstein
May 27, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Surfaces/Plane.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Plane.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Initialization

Intercept Form

Plane[a_,b_,c_]:=Polygon[{{a,0,0},{0,b,0},{0,0,c}}]

<<ImplicitPlot3D`

pts={3,2,1};
Show[Graphics3D[{
Function[{a,b,c},ImplicitPlot3D[x/a+y/b+z/c==1,{x,0,a},{y,0,b},{z,0,c},DisplayFunction->Identity][[1]]]@@pts,
PointSize[.03],
Point/@(IdentityMatrix[3]pts)
}]]

pts={3,2,1};
Show[Graphics3D[{
Plane@@pts,
PointSize[.03],
Point/@(IdentityMatrix[3]pts)
}]]

a-b-c-d-Form

Plane[a_,b_,c_,d_]:=Plane[-d/a,-d/b,-d/c]

Show[Graphics3D[{
Plane[1,2,3,4]
}]]

Three-Point Form

ReplacePart[Table[a[i,j],{i,3},{j,3}],1,Array[{#,1}&,3]]

plane[xx:{x1_,x2_,x3_}]:=Module[{a1,a2,a3},
a1=a2=a3=xx;
a1[[All,1]]=1;
a2[[All,2]]=1;
a3[[All,3]]=1;
{x,y,z}.(Det/@{a1,a2,a3})-Det[xx]
]

plane[{{1,0,1},{0,1,0},{1,1,0}}]

(1-y-z/.Thread[{x,y,z}->#])&/@{{1,0,1},{0,1,0},{1,1,0}}

plane[Array[xx,{3,3}]]

Simplify[(%/.Thread[{x,y,z}->#])&/@Array[xx,{3,3}]]

Properties

surf[a_,b_,c_][u_,v_]:={u,v,a u+b v+c}

ParametricPlot3D[surf[Sequence@@#][u,v],{u,-1,1},{v,-1,1}]&/@{{0,0,0},{1,0,0},{0,1,0},{0,0,1}}

Curvatures[surf[a,b,c][u,v],{u,v}]