Spheric Section

Author

Eric W. Weisstein
January 29, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SolidGeometry/SphericSection.nb.

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

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

Code

SphericSection[{a_,b_,c_,d_},r_:1,opts___]:=Module[{curve,eps=10^-5},
Cases[ParametricPlot3D[{x,-(((b^2) d+a (b^2) x+#c (Sqrt[-(d^2)+(c^2) (r^2)-2 a d x-(a^2) (x^2)-(c^2) (x^2)-(b^2) (-(r^2)+(x^2))]) Abs[b])/(b ((b^2)+(c^2)))),((-c d-a c x+#(Sqrt[-(d^2)+(c^2) (r^2)-2 a d x-(a^2) (x^2)-(c^2) (x^2)-(b^2) (-(r^2)+(x^2))]) Abs[b])/((b^2)+(c^2)))}&/@{-1,1},{x,((-2 a d-(Sqrt[4 (a^2) (d^2)-4 (-(a^2)-(b^2)-(c^2)) (-(d^2)+(b^2) (r^2)+(c^2) (r^2))]))/(2 ((a^2)+(b^2)+(c^2))))+eps,((-2 a d+(Sqrt[4 (a^2) (d^2)-4 (-(a^2)-(b^2)-(c^2)) (-(d^2)+(b^2) (r^2)+(c^2) (r^2))]))/(2 ((a^2)+(b^2)+(c^2))))-eps},opts],_Line,{1,∞}]
]

Equations

Refine[Solve[{x^2+y^2+z^2==r^2,a x+b y +c z+d==0},{y,z}]//Simplify,(a|b|c|d|r)∈Reals]

Simplify[Solve[-(d^2)+(c^2) (r^2)-2 a d x-(a^2) (x^2)-(c^2) (x^2)-(b^2) (-(r^2)+(x^2))==0,x]]

Figure

Graphics3D[{Sphere[],
Red,Thickness[.01],
SphericSection/@{{0,1,0,0},{0,1,.2,.95},{1,1,.2,-.95},{0,0.0001,1,0}}
},Axes->False]