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Linear interpolation

Linear interpolation is a science 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 Linear interpolation rather than just read about it. In short: In mathematics, linear interpolation (sometimes lerp) is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points. Linear interpolation between two known points If the two known points are given by the coordinates ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} and ( x 1 , y 1 ) {\displaystyle (x_{1},y_{1})} , the linear interpolant is the st…

Linear interpolation — main illustration
Linear interpolation — illustration

Key takeaways

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

Reference excerpt

In mathematics, linear interpolation (sometimes lerp) is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points.

Linear interpolation between two known points

If the two known points are given by the coordinates ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} and ( x 1 , y 1 ) {\displaystyle (x_{1},y_{1})} , the linear interpolant is the straight line between these points. For a value x {\displaystyle x} in the interval ( x 0 , x 1 ) {\displaystyle (x_{0},x_{1})} , the value y {\displaystyle y} along the straight line is given from the equation of slopes

y − y 0 x − x 0 = y 1 − y 0 x 1 − x 0 , {\displaystyle {\frac {y-y_{0}}{x-x_{0}}}={\frac {y_{1}-y_{0}}{x_{1}-x_{0}}},}

which can be derived geometrically from the figure on the right. It is a special case of polynomial interpolation with n = 1 {\displaystyle n=1} . Solving this equation for y {\displaystyle y} , which is the unknown value at x {\displaystyle x} , gives

… excerpt ends here. Continue reading the full article.

Illustrations

Linear interpolation: Given the two red points, the blue line is the linear interpolant between the points, and the value y at x may be found by linear interpolation.
Given the two red points, the blue line is the linear interpolant between the points, and the value y at x may be found by linear interpolation.
Linear interpolation: In this geometric visualisation, the value at the green circle multiplied by the horizontal distance between the red and blue circles is equal to the sum of the value at the red circle multiplied by the horizontal distance between the green and blue circles, and the value at the blue circle multiplied by the horizontal distance between the green and red circles.
In this geometric visualisation, the value at the green circle multiplied by the horizontal distance between the red and blue circles is equal to the sum of the value at the red circle multiplied by the horizontal distance between the green and blue circles, and the value at the blue circle multiplied by the horizontal distance between the green and red circles.
Linear interpolation: Linear interpolation on a data set (red points) consists of pieces of linear interpolants (blue lines).
Linear interpolation on a data set (red points) consists of pieces of linear interpolants (blue lines).
Linear interpolation: Comparison of linear and bilinear interpolation some 1- and 2-dimensional interpolations. Black and red/yellow/green/blue dots correspond to the interpolated point and neighbouring samples, respectively. Their heights above the ground correspond to their values.
Comparison of linear and bilinear interpolation some 1- and 2-dimensional interpolations. Black and red/yellow/green/blue dots correspond to the interpolated point and neighbouring samples, respectively. Their heights above the ground correspond to their values.
Linear interpolation: Example of bilinear interpolation on the unit square with the z values 0, 1, 1, and 0.5 as indicated. Interpolated values in between are represented by colour.
Example of bilinear interpolation on the unit square with the z values 0, 1, 1, and 0.5 as indicated. Interpolated values in between are represented by colour.

Worked examples

Example 1 — a first encounter with Linear interpolation

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

In research
Linear interpolation appears in 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 Linear interpolation 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
Linear interpolation is common in secondary-school and first-year university syllabi. It links to neighbouring topics Interpolation, so understanding it makes those chapters shorter.
In everyday life
Look for Linear interpolation 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 Linear interpolation in 20 minutes

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

Frequently asked questions

What is Linear interpolation in simple terms?

In mathematics, linear interpolation (sometimes lerp) is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points. Linear interpolation between two known points If the two known points are given by the coordinates ( x 0…

Why does Linear interpolation matter?

Because it connects several 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 Linear interpolation?

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 Linear interpolation.

Tags

  • Interpolation

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