Geometry

a₁ a₁x arbitrary point asymptotic ax+by+cz axiom system axis b₁ b₁y c₁ Cartesian coordinates Cartesian model centre circle coefficients congruent conic section conjugate constructed convex coordinate axes coordinate system curve d₁ dihedral angles distance Dupin indicatrix dv² edges ellipse equal Euclidean geometry Euler characteristic f₁ f₂ face angles Find Form the equation formulas fundamental form Gaussian curvature geodesic given point half-line half-plane Hence hyperbola k₁ length line segment linear Lobachevsky locus of points normal curvature obtain osculating plane P₁ parabola paraboloid parametric parametric equations perpendicular points of intersection polar polygon polyhedron projective transformation proof prove second fundamental form Show sides specified sphere straight line passing straight lines parallel surface tangent plane theorem three points triangle ABC vector vertex vertices x-axis x₁ xy-plane y₁ yz-plane z-axis zero