Cusp (singularity)

Consider a smooth real-valued function of two variables, say where and are real numbers. So is a function from the plane to the line. The space of all such smooth functions is acted upon by the group of diffeomorphisms of the plane and the diffeomorphisms of the line, i.e. diffeomorphic changes of coordinate in both the source and the target. This action splits the whole function space up into equivalence classes, i.e. orbits of the group action. One such family of equivalence classes is denoted by where is a non-negative integer. A function is said to be of type if it lies in the orbit of x^2 \pm y^{k+1}, i.e. there exists a diffeomorphic change of coordinate in source and target which takes into one of these forms. These simple forms x^2 \pm y^{k+1} are said to give normal forms for the type -singularities. Notice that the {{tmath|A_{2n}^+}} are the same as the {{tmath|A_{2n}^-}} since the diffeomorphic change of coordinate in the source takes x^2 + y^{k+1} to x^2 - y^{2n+1}. So we can drop the ± from {{tmath|A_{2n}^\pm}} notation. The cusps are then given by the zero-level-sets of the representatives of the {{tmath|A_{2n} }} equivalence classes, where is an integer. ==Examples==

y^2=x^3 • An ordinary cusp is given by x^2-y^3=0, i.e. the zero-level-set of a type -singularity. Let be a smooth function of and and assume, for simplicity, that . Then a type -singularity of at can be characterised by: • Having a degenerate quadratic part, i.e. the quadratic terms in the Taylor series of form a perfect square, say , where is linear in and , and • does not divide the cubic terms in the Taylor series of . • A rhamphoid cusp () originally denoted a cusp such that both branches are on the same side of the tangent, such as for the curve of equation (x-y^2)^2-y^5=0. As such a singularity is in the same differential class as the cusp of equation x^2-y^5=0, which is a singularity of type , the term has been extended to all such singularities. These cusps are non-generic as caustics and wave fronts. The rhamphoid cusp and the ordinary cusp are non-diffeomorphic. A parametric form is x = t^2,\, y = a t^4 + t^5,\, a \neq 0. For a type -singularity we need to have a degenerate quadratic part (this gives type ), that does divide the cubic terms (this gives type ), another divisibility condition (giving type ), and a final non-divisibility condition (giving type exactly ). To see where these extra divisibility conditions come from, assume that has a degenerate quadratic part and that divides the cubic terms. It follows that the third order taylor series of is given by L^2 \pm LQ, where is quadratic in and . We can complete the square to show that L^2 \pm LQ = (L \pm Q/2)^2 - Q^4/4. We can now make a diffeomorphic change of variable (in this case we simply substitute polynomials with linearly independent linear parts) so that (L \pm Q/2)^2 - Q^4/4 \to x_1^2 + P_1 where is quartic (order four) in and . The divisibility condition for type is that divides . If does not divide then we have type exactly (the zero-level-set here is a tacnode). If divides we complete the square on x_1^2 + P_1 and change coordinates so that we have x_2^2 + P_2 where is quintic (order five) in and . If does not divide then we have exactly type , i.e. the zero-level-set will be a rhamphoid cusp. ==Applications==

of light rays in the bottom of a teacup. Cusps appear naturally when projecting into a plane a smooth curve in three-dimensional Euclidean space. In general, such a projection is a curve whose singularities are self-crossing points and ordinary cusps. Self-crossing points appear when two different points of the curves have the same projection. Ordinary cusps appear when the tangent to the curve is parallel to the direction of projection (that is when the tangent projects on a single point). More complicated singularities occur when several phenomena occur simultaneously. For example, rhamphoid cusps occur for inflection points (and for undulation points) for which the tangent is parallel to the direction of projection. In many cases, and typically in computer vision and computer graphics, the curve that is projected is the curve of the critical points of the restriction to a (smooth) spatial object of the projection. A cusp appears thus as a singularity of the contour of the image of the object (vision) or of its shadow (computer graphics). Caustics and wave fronts are other examples of curves having cusps that are visible in the real world. ==See also==