frenetellipse.mws a calculus max/min distance problem leads to counting the normals describing an ellipse aside: the algebra confirming this remark visualizing the problem loading the vec_calc package annoying warning animated ellipse with evolute the procedure ellipsevolute (and bonus procedure ellipsevolute_withcircle) evolute inside or outside of the ellipse normal lines which pass through a given point visualize it making this into a procedure: ellipser(b,r,theta) and the multivalued distance functions: ellipsedistance(b,r,theta) and ellipsedistance_min_max(b,r,theta) plotting the multivalued distance functions (time consuming) explicit equations for the evolute of an ellipse number of normals passing through a given point: root analysis the discriminant of a quartic (general theory) discriminant sign cases let's backup: discriminant background without the old books Discriminant in terms of u,v; (never give up!) referee sum of squares representation the hyperbolas of Apollonius the extended procedure ellipser_hyp(b,r,theta) setting up the diagram new triangle figure intersection of the ellipse and hyperbola: nonparametric approach (hindsight) properties of the hyperbola the extended procedure ellipser_hyp_center(b,r,theta) and its animation final frame export bonus: evolute as the envelope of normals and the locus of singularities of the family of parallels (graphical) ellipsevolute_normals(b,n) notickmark diagram ellipsevolute_parallels(b,s) ellipsevolute_parallels_movie(b,sfinal) bonus: evolute as the curve of limiting intersections of successive normals (symbolic) following up by hand references and web links generalizations?