Английская Википедия:Cusp (singularity)
Шаблон:Short description Шаблон:No footnotes
In mathematics, a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point must reverse direction. A typical example is given in the figure. A cusp is thus a type of singular point of a curve.
For a plane curve defined by an analytic, parametric equation
- <math>\begin{align}
x &= f(t)\\ y &= g(t), \end{align} </math> a cusp is a point where both derivatives of Шаблон:Math and Шаблон:Math are zero, and the directional derivative, in the direction of the tangent, changes sign (the direction of the tangent is the direction of the slope <math> \lim (g'(t)/f'(t))</math>). Cusps are local singularities in the sense that they involve only one value of the parameter Шаблон:Math, in contrast to self-intersection points that involve more than one value. In some contexts, the condition on the directional derivative may be omitted, although, in this case, the singularity may look like a regular point.
For a curve defined by an implicit equation
- <math>F(x,y) = 0,</math>
which is smooth, cusps are points where the terms of lowest degree of the Taylor expansion of Шаблон:Math are a power of a linear polynomial; however, not all singular points that have this property are cusps. The theory of Puiseux series implies that, if Шаблон:Math is an analytic function (for example a polynomial), a linear change of coordinates allows the curve to be parametrized, in a neighborhood of the cusp, as
- <math>\begin{align}
x &= at^m\\ y &= S(t), \end{align} </math> where Шаблон:Math is a real number, Шаблон:Math is a positive even integer, and Шаблон:Math is a power series of order Шаблон:Math (degree of the nonzero term of the lowest degree) larger than Шаблон:Math. The number Шаблон:Math is sometimes called the order or the multiplicity of the cusp, and is equal to the degree of the nonzero part of lowest degree of Шаблон:Math. In some contexts, the definition of a cusp is restricted to the case of cusps of order two—that is, the case where Шаблон:Math.
The definitions for plane curves and implicitly-defined curves have been generalized by René Thom and Vladimir Arnold to curves defined by differentiable functions: a curve has a cusp at a point if there is a diffeomorphism of a neighborhood of the point in the ambient space, which maps the curve onto one of the above-defined cusps.
Classification in differential geometry
Consider a smooth real-valued function of two variables, say Шаблон:Math where Шаблон:Mvar and Шаблон:Mvar are real numbers. So Шаблон:Mvar 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 Шаблон:Tmath where Шаблон:Mvar is a non-negative integer. A function Шаблон:Mvar is said to be of type Шаблон:Tmath if it lies in the orbit of <math>x^2 \pm y^{k+1},</math> i.e. there exists a diffeomorphic change of coordinate in source and target which takes Шаблон:Mvar into one of these forms. These simple forms <math>x^2 \pm y^{k+1}</math> are said to give normal forms for the type Шаблон:Tmath-singularities. Notice that the Шаблон:Tmath are the same as the Шаблон:Tmath since the diffeomorphic change of coordinate Шаблон:Tmath in the source takes <math>x^2 + y^{k+1}</math> to <math>x^2 - y^{2n+1}.</math> So we can drop the ± from Шаблон:Tmath notation.
The cusps are then given by the zero-level-sets of the representatives of the Шаблон:Tmath equivalence classes, where Шаблон:Math is an integer. Шаблон:Citation needed
Examples
- An ordinary cusp is given by <math>x^2-y^3=0,</math> i.e. the zero-level-set of a type Шаблон:Math-singularity. Let Шаблон:Math be a smooth function of Шаблон:Mvar and Шаблон:Mvar and assume, for simplicity, that Шаблон:Math. Then a type Шаблон:Math-singularity of Шаблон:Mvar at Шаблон:Math can be characterised by:
- Having a degenerate quadratic part, i.e. the quadratic terms in the Taylor series of Шаблон:Mvar form a perfect square, say Шаблон:Math, where Шаблон:Math is linear in Шаблон:Mvar and Шаблон:Mvar, and
- Шаблон:Math does not divide the cubic terms in the Taylor series of Шаблон:Math.
- A rhamphoid cusp (Шаблон:Ety) denoted originally a cusp such that both branches are on the same side of the tangent, such as for the curve of equation <math>x^2-x^4-y^5=0.</math> As such a singularity is in the same differential class as the cusp of equation <math>x^2-y^5=0,</math> which is a singularity of type Шаблон:Math, 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 <math>x = t^2,\, y = a x^4 + x^5.</math>
For a type Шаблон:Math-singularity we need Шаблон:Mvar to have a degenerate quadratic part (this gives type Шаблон:Math), that Шаблон:Mvar does divide the cubic terms (this gives type Шаблон:Math), another divisibility condition (giving type Шаблон:Math), and a final non-divisibility condition (giving type exactly Шаблон:Math).
To see where these extra divisibility conditions come from, assume that Шаблон:Mvar has a degenerate quadratic part Шаблон:Math and that Шаблон:Mvar divides the cubic terms. It follows that the third order taylor series of Шаблон:Mvar is given by <math>L^2 \pm LQ,</math> where Шаблон:Mvar is quadratic in Шаблон:Mvar and Шаблон:Mvar. We can complete the square to show that <math>L^2 \pm LQ = (L \pm Q/2)^2 - Q^4/4.</math> We can now make a diffeomorphic change of variable (in this case we simply substitute polynomials with linearly independent linear parts) so that <math>(L \pm Q/2)^2 - Q^4/4 \to x_1^2 + P_1</math> where Шаблон:Math is quartic (order four) in Шаблон:Math and Шаблон:Math. The divisibility condition for type Шаблон:Math is that Шаблон:Math divides Шаблон:Math. If Шаблон:Math does not divide Шаблон:Math then we have type exactly Шаблон:Math (the zero-level-set here is a tacnode). If Шаблон:Math divides Шаблон:Math we complete the square on <math>x_1^2 + P_1</math> and change coordinates so that we have <math>x_2^2 + P_2</math> where Шаблон:Math is quintic (order five) in Шаблон:Math and Шаблон:Math. If Шаблон:Math does not divide Шаблон:Math then we have exactly type Шаблон:Math, i.e. the zero-level-set will be a rhamphoid cusp.
Applications
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
References
External links
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