Tacnode

Key takeaways

Tacnode belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Tacnode to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Tacnode from memory before moving on to harder problems.

Reference excerpt

In classical algebraic geometry, a tacnode (also called a point of osculation or double cusp) is a kind of singular point of a curve. It is defined as a point where two (or more) osculating circles to the curve at that point are tangent. This means that two branches of the curve have ordinary tangency at the double point. The canonical example is

y 2 − x 4 = 0. {\displaystyle y^{2}-x^{4}=0.}

A tacnode of an arbitrary curve may then be defined from this example, as a point of self-tangency locally diffeomorphic to the point at the origin of this curve. Another example of a tacnode is given by the links curve shown in the figure, with equation

( x 2 + y 2 − 3 x ) 2 − 4 x 2 ( 2 − x ) = 0. {\displaystyle (x^{2}+y^{2}-3x)^{2}-4x^{2}(2-x)=0.}

More general background Consider a smooth real-valued function of two variables, say f (x, y) where x and y are real numbers. So f 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 ⁠ A k ± , {\displaystyle A_{k}^{\pm },} ⁠ where k is a non-negative integer. This notation was introduced by V. I. Arnold. A function f is said to be of type ⁠ A k ± {\displaystyle A_{k}^{\pm }} ⁠ if it lies in the orbit of x 2 ± y k + 1 , {\displaystyle x^{2}\pm y^{k+1},} i.e. there exists a diffeomorphic change of coordinate in source and target which takes f into one of these forms. These simple forms x 2 ± y k + 1 {\displaystyle x^{2}\pm y^{k+1}} are said to give normal forms for the type ⁠ A k ± {\displaystyle A_{k}^{\pm }} ⁠-singularities. A curve with equation f = 0 will have a tacnode, say at the origin, if and only if f has a type ⁠ A 3 − {\displaystyle A_{3}^{-}} ⁠-singularity at the origin. Notice that a node ( x 2 − y 2 = 0 ) {\displaystyle (x^{2}-y^{2}=0)} corresponds to a type ⁠ A 1 − {\displaystyle A_{1}^{-}} ⁠-singularity. A tacnode corresponds to a type ⁠ A 3 − {\displaystyle A_{3}^{-}} ⁠-singularity. In fact each type ⁠ A 2 n + 1 − {\displaystyle A_{2n+1}^{-}} ⁠-singularity, where n ≥ 0 is an integer, corresponds to a curve with self-intersection. As n increases, the order of self-intersection increases: transverse crossing, ordinary tangency, etc. The type ⁠ A 2 n + 1 + {\displaystyle A_{2n+1}^{+}} ⁠-singularities are of no interest over the real numbers: they all give an isolated point. Over the complex numbers, type ⁠ A 2 n + 1 + {\displaystyle A_{2n+1}^{+}} ⁠-singularities and type ⁠ A 2 n + 1 − {\displaystyle A_{2n+1}^{-}} ⁠-singularities are equivalent: (x, y) → (x, iy) gives the required diffeomorphism of the normal forms.

Illustrations

Tacnode: A tacnode at the origin of the curve defined by

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{\displaystyle (x^{2}+y^{2}-3x)^{2}-4x^{2}(2-x)=0.} — A tacnode at the origin of the curve defined by ( x 2 + y 2 − 3 x ) 2 − 4 x 2 ( 2 − x ) = 0. {\displaystyle (x^{2}+y^{2}-3x)^{2}-4x^{2}(2-x)=0.}

Worked examples

Example 1 — a first encounter with Tacnode

Start with the simplest possible case. Write down what Tacnode claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Tacnode before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Tacnode ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Tacnode

In research: Tacnode appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Tacnode in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Tacnode is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic curves, Curves, Singularity theory, so understanding it makes those chapters shorter.
In everyday life: Look for Tacnode outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Tacnode in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Tacnode means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Tacnode out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Tacnode in simple terms?

In classical algebraic geometry, a tacnode (also called a point of osculation or double cusp) is a kind of singular point of a curve. It is defined as a point where two (or more) osculating circles to the curve at that point are tangent.

Why does Tacnode matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Tacnode?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Tacnode.