mathematics

Trident curve

Trident curve is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Trident curve rather than just read about it. In short: In mathematics, a trident curve (also trident of Newton or parabola of Descartes) is any member of the family of curves that have the formula: x y + a x 3 + b x 2 + c x = d {\displaystyle xy+ax^{3}+bx^{2}+cx=d} . Trident curves are cubic plane curves with an ordinary double point in the real projective plane at x = 0 {\displaystyle x=0} , y = 1 {\displaystyle y=1} , z = 0 {\displaystyle z=0} .

Key takeaways

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

Reference excerpt

In mathematics, a trident curve (also trident of Newton or parabola of Descartes) is any member of the family of curves that have the formula:

x y + a x 3 + b x 2 + c x = d {\displaystyle xy+ax^{3}+bx^{2}+cx=d} .

Trident curves are cubic plane curves with an ordinary double point in the real projective plane at x = 0 {\displaystyle x=0} , y = 1 {\displaystyle y=1} , z = 0 {\displaystyle z=0} . If we substitute x = x / z {\displaystyle x=x/z} and y = 1 / z {\displaystyle y=1/z} into the equation of the trident curve, we get

a x 3 + b x 2 z + c x z 2 + x z = d z 3 , {\displaystyle ax^{3}+bx^{2}z+cxz^{2}+xz=dz^{3},}

which has an ordinary double point at the origin. Trident curves are therefore rational plane algebraic curves of genus zero. Solving for y {\displaystyle y} , we get

y = d x − a x 2 − b x − c {\displaystyle y={\frac {d}{x}}-ax^{2}-bx-c} . Solving for x {\displaystyle x} , we get

x = d − a x 3 − b x 2 − c x y {\displaystyle x={\frac {d-ax^{3}-bx^{2}-cx}{y}}} .

References

External links O'Connor, John J.; Robertson, Edmund F., "Trident of Newton", MacTutor History of Mathematics Archive, University of St Andrews

Illustrations

Worked examples

Example 1 — a first encounter with Trident curve

Start with the simplest possible case. Write down what Trident curve 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 Trident curve 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 Trident curve 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 Trident curve

In research: Trident curve 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 Trident curve 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: Trident curve is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic geometry stubs, Cubic curves, so understanding it makes those chapters shorter.
In everyday life: Look for Trident curve 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.

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How to study Trident curve in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Trident curve 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 Trident curve out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Trident curve in simple terms?

In mathematics, a trident curve (also trident of Newton or parabola of Descartes) is any member of the family of curves that have the formula: x y + a x 3 + b x 2 + c x = d {\displaystyle xy+ax^{3}+bx^{2}+cx=d} . Trident curves are cubic plane curves with an ordinary double point in the real projec…

Why does Trident curve 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 Trident curve?

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 Trident curve.