partial class Examples
{
  public static Result PrincipalCurvature(RhinoDoc doc)
  {
    ObjRef obj_ref;
    var rc = RhinoGet.GetOneObject("Select surface for curvature measurement", true,
      ObjectType.Surface, out obj_ref);
    if (rc != Result.Success)
      return rc;
    var surface = obj_ref.Surface();

    var gp = new Rhino.Input.Custom.GetPoint();
    gp.SetCommandPrompt("Select point on surface for curvature measurement");
    gp.Constrain(surface, false);
    gp.Get();
    if (gp.CommandResult() != Result.Success)
      return gp.CommandResult();
    var point_on_surface = gp.Point();

    double u, v;
    if (!surface.ClosestPoint(point_on_surface, out u, out v))
      return Result.Failure;

    var surface_curvature = surface.CurvatureAt(u, v);
    if (surface_curvature == null)
      return Result.Failure;

    RhinoApp.WriteLine("Surface curvature evaluation at parameter: ({0}, {1})", u, v);

    RhinoApp.WriteLine("  3-D Point: ({0}, {1}, {2})",
      surface_curvature.Point.X,
      surface_curvature.Point.Y,
      surface_curvature.Point.Z);

    RhinoApp.WriteLine("  3-D Normal: ({0}, {1}, {2})",
      surface_curvature.Normal.X,
      surface_curvature.Normal.Y,
      surface_curvature.Normal.Z);

    RhinoApp.WriteLine(string.Format("  Maximum principal curvature: {0} ({1}, {2}, {3})",
      surface_curvature.Kappa(0),
      surface_curvature.Direction(0).X,
      surface_curvature.Direction(0).Y,
      surface_curvature.Direction(0).Z));

    RhinoApp.WriteLine(string.Format("  Minimum principal curvature: {0} ({1}, {2}, {3})",
      surface_curvature.Kappa(1),
      surface_curvature.Direction(1).X,
      surface_curvature.Direction(1).Y,
      surface_curvature.Direction(1).Z));

    RhinoApp.WriteLine("  Gaussian curvature: {0}", surface_curvature.Gaussian);
    RhinoApp.WriteLine("  Mean curvature: {0}", surface_curvature.Mean);

    return Result.Success;
  }
}

Partial Friend Class Examples
  Public Shared Function PrincipalCurvature(ByVal doc As RhinoDoc) As Result
	Dim obj_ref As ObjRef = Nothing
	Dim rc = RhinoGet.GetOneObject("Select surface for curvature measurement", True, ObjectType.Surface, obj_ref)
	If rc IsNot Result.Success Then
	  Return rc
	End If
	Dim surface = obj_ref.Surface()

	Dim gp = New Rhino.Input.Custom.GetPoint()
	gp.SetCommandPrompt("Select point on surface for curvature measurement")
	gp.Constrain(surface, False)
	gp.Get()
	If gp.CommandResult() <> Result.Success Then
	  Return gp.CommandResult()
	End If
	Dim point_on_surface = gp.Point()

	Dim u As Double = Nothing, v As Double = Nothing
	If Not surface.ClosestPoint(point_on_surface, u, v) Then
	  Return Result.Failure
	End If

	Dim surface_curvature = surface.CurvatureAt(u, v)
	If surface_curvature Is Nothing Then
	  Return Result.Failure
	End If

	RhinoApp.WriteLine("Surface curvature evaluation at parameter: ({0}, {1})", u, v)

	RhinoApp.WriteLine("  3-D Point: ({0}, {1}, {2})", surface_curvature.Point.X, surface_curvature.Point.Y, surface_curvature.Point.Z)

	RhinoApp.WriteLine("  3-D Normal: ({0}, {1}, {2})", surface_curvature.Normal.X, surface_curvature.Normal.Y, surface_curvature.Normal.Z)

	RhinoApp.WriteLine(String.Format("  Maximum principal curvature: {0} ({1}, {2}, {3})", surface_curvature.Kappa(0), surface_curvature.Direction(0).X, surface_curvature.Direction(0).Y, surface_curvature.Direction(0).Z))

	RhinoApp.WriteLine(String.Format("  Minimum principal curvature: {0} ({1}, {2}, {3})", surface_curvature.Kappa(1), surface_curvature.Direction(1).X, surface_curvature.Direction(1).Y, surface_curvature.Direction(1).Z))

	RhinoApp.WriteLine("  Gaussian curvature: {0}", surface_curvature.Gaussian)
	RhinoApp.WriteLine("  Mean curvature: {0}", surface_curvature.Mean)

	Return Result.Success
  End Function
End Class

import rhinoscriptsyntax as rs

surface_id,_,_,_,_,_ = rs.GetSurfaceObject("Select surface for curvature measurement")
point = rs.GetPointOnSurface(surface_id, "Select point on surface for curvature measurement")
u,v = rs.SurfaceClosestPoint(surface_id, point)

#point, normal, kappa_u, direction_u, kappa_v, direction_v, gaussian, mean =
surface_curvature = rs.SurfaceCurvature(surface_id, (u,v))

point, normal, kappa_u, direction_u, kappa_v, direction_v, gaussian, mean = surface_curvature

print "Surface curvature evaluation at parameter: ({0}, {1})".format(u,v)

print "  3-D Point: ({0}, {1}, {2})".format(point.X, point.Y, point.Z)

print "  3-D Normal: ({0}, {1}, {2})".format(normal.X, normal.Y, normal.Z)

print "  Maximum principal curvature: {0} ({1}, {2}, {3})".format(kappa_u, direction_u.X, direction_u.Y, direction_u.Z)

print "  Minimum principal curvature: {0} ({1}, {2}, {3})".format(kappa_v, direction_v.X, direction_v.Y, direction_v.Z)

print "  Gaussian curvature: {0}".format(gaussian)
print "  Mean curvature: {0}".format(mean)