   Extreme Values of a Closed Surface / Normal Vector
  Hello folks, I have a question:As fas as I know, if you have an ellipsoid and a point p lying on the surface of that ellipsoid, then point p corresponds to the extreme point on the surface in the direction of the surface normal vector in point p.Which properties does a surface have to fulfill in order for this rule to be true?   
Quote:Original post by DesperadoHello folks, I have a question:As fas as I know, if you have an ellipsoid and a point p lying on the surface of that ellipsoid, then point p corresponds to the extreme point on the surface in the direction of the surface normal vector in point p.Which properties does a surface have to fulfill in order for this rule to be true?Convexity.   ;  Thanks. If you could also provide me with a reference that formulates it that way, I would be completely satisfied.   ;  Sorry for being that late:Computational Geometry: Algorithms and ApplicationsThird Edition (March 2008)Mark de Berg, TU Eindhoven (the Netherlands)Otfried Cheong, KAIST (Korea)Marc van Kreveld, Mark Overmars, Utrecht University (the Netherlands)   ;  Very well, thank you.One more question:I have a bending operation that makes a circular bend on an ellipsoid along it's longest axis (the axis gets bent to a circle segment and the circle radius determines the curvature of the bend). This, of course, makes the ellipsoid concave. Is it still possible to efficiently calculate the extreme point into a direction for such a shape?Thanks   ;Quote:Original post by DesperadoVery well, thank you.One more question:I have a bending operation that makes a circular bend on an ellipsoid along it's longest axis (the axis gets bent to a circle segment and the circle radius determines the curvature of the bend). This, of course, makes the ellipsoid concave. Is it still possible to efficiently calculate the extreme point into a direction for such a shape?ThanksYou can always use a brute force method.Sorry for the reference. The right one is:Computational Geometry in C, Second EditionJoseph O'RourkeCambridge PressThere's a subsection devoted to that specific question:Chap 7.10: Extremal polytope Queries p. 280Reading the introduction of the subsection, nothing is said about the convexity (or non convexity) of the polytope. The method exposed has an O(logN) complexity.   