Smoothstep

Key takeaways

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

Reference excerpt

Smoothstep is a family of sigmoid-like interpolation and clamping functions commonly used in computer graphics, video game engines, and machine learning. The function depends on three parameters, the input x, the "left edge" and the "right edge", with the left edge being assumed smaller than the right edge. The function receives a real number x as an argument. It returns 0 if x is less than or equal to the left edge and 1 if x is greater than or equal to the right edge. Otherwise, it smoothly interpolates, using Hermite interpolation, and returns a value between 0 and 1. The slope of the smoothstep function is zero at both edges. This is convenient for creating a sequence of transitions using smoothstep to interpolate each segment as an alternative to using more sophisticated or expensive interpolation techniques. In HLSL and GLSL, smoothstep implements the S 1 ⁡ ( x ) {\displaystyle \operatorname {S} _{1}(x)} , the cubic Hermite interpolation after clamping:

smoothstep ⁡ ( x ) = S 1 ( x ) = { 0 , x ≤ 0 3 x 2 − 2 x 3 , 0 ≤ x ≤ 1 1 , 1 ≤ x {\displaystyle \operatorname {smoothstep} (x)=S_{1}(x)={\begin{cases}0,&x\leq 0\\3x^{2}-2x^{3},&0\leq x\leq 1\\1,&1\leq x\\\end{cases}}}

Assuming that the left edge is 0, the right edge is 1, with the transition between edges taking place where 0 ≤ x ≤ 1. A modified C/C++ example implementation provided by AMD follows.

The general form for smoothstep, again assuming the left edge is 0 and right edge is 1, is

S n ⁡ ( x ) = { 0 , if x ≤ 0 x n + 1 ∑ k = 0 n ( n + k k ) ( 2 n + 1 n − k ) ( − x ) k , if 0 ≤ x ≤ 1 1 , if 1 ≤ x {\displaystyle \operatorname {S} _{n}(x)={\begin{cases}0,&{\text{if }}x\leq 0\\x^{n+1}\sum _{k=0}^{n}{\binom {n+k}{k}}{\binom {2n+1}{n-k}}(-x)^{k},&{\text{if }}0\leq x\leq 1\\1,&{\text{if }}1\leq x\\\end{cases}}}

S 0 ⁡ ( x ) {\displaystyle \operatorname {S} _{0}(x)} is identical to the clamping function:

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Smoothstep

Start with the simplest possible case. Write down what Smoothstep claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In computer science, 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 Smoothstep 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 Smoothstep 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 Smoothstep

In research: Smoothstep appears in computer science 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 Smoothstep 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: Smoothstep is common in secondary-school and first-year university syllabi. It links to neighbouring topics Computer graphics algorithms, Sigmoid functions, so understanding it makes those chapters shorter.
In everyday life: Look for Smoothstep 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 Smoothstep in 20 minutes

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

Frequently asked questions

What is Smoothstep in simple terms?

Smoothstep is a family of sigmoid-like interpolation and clamping functions commonly used in computer graphics, video game engines, and machine learning. The function depends on three parameters, the input x, the "left edge" and the "right edge", with the left edge being assumed smaller than the ri…

Why does Smoothstep matter?

Because it connects several computer science 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 Smoothstep?

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 Smoothstep.