# Interpolation and Curve Fitting in MATLAB: Video Tutorial

URL: https://yarpiz.com/interpolation-video-tutorial-yptnm190520s07
Author: Mostapha Kalami Heris
Published: 2019-07-03
Updated: 2026-10-30
Topics: Courses

> Compare polynomial fitting, grid interpolation, Fourier interpolation, splines and scattered-data methods in MATLAB, with practical checks for noisy data and extrapolation.

This video tutorial reviews interpolation and curve fitting and implements them in MATLAB. It is part of the [Numerical Computations in MATLAB](https://yarpiz.com/numerical-computations-in-matlab-video-tutorial-572) course.

## Interpolation and curve fitting

Both methods start from a set of data points $(x_i, y_i)$ and build a model that gives values between them. They differ in one important way:

- **Interpolation** builds a model that passes exactly through every data point, for example a straight line or a cubic piece between each pair of neighbors.
- **Curve fitting** chooses a simple model, such as a polynomial of degree $n$, that stays as close as possible to the points. With noisy data it does not pass through them. `polyfit` uses least squares: it picks the coefficients that minimize the sum of squared errors $\sum_i \left(y_i - p(x_i)\right)^2$.

The five videos move from one input variable to two and three, and from evenly spaced data to scattered points.

## Watch online

The tutorial has five sections on YouTube. They are embedded below as a playlist.

::: video {playlist="PLgH3sgdvgO4T23TWF9Q0-5DSKV_7H9OTj" youtube="9Dui-ExAEEc" count="5" title="Interpolation in MATLAB: Video Tutorial"}
:::

## Video sections

### 1. Polynomial fitting with `polyfit`

Polynomial fitting using `polyfit` ([YouTube](https://youtu.be/9Dui-ExAEEc)). The function `polyfit(x, y, n)` takes the data and a degree $n$ and returns the coefficients of the polynomial. The video starts with three points of the known model $x^2 - 3x + 5$, and a fit of degree 2 returns exactly `[1 -3 5]`.

It then draws 100 random points between $-1$ and $5$ and adds normally distributed noise. The fitted coefficients stay close to 1, $-3$ and 5, so `polyfit` recovers the model behind the noisy data. `polyval` evaluates the fitted polynomial for the plot. A degree of 1 gives the best straight line, and a degree of 3 adds a cubic term that the data does not need.

```matlab {title="Fit and evaluate a quadratic polynomial"}
p = polyfit(x, y, 2);
yy = polyval(p, xx);
```

Finally, the second output `S` of `polyfit` is passed to `polyval` to get prediction error estimates `delta` at each point. The video plots `yy + 2*delta` and `yy - 2*delta` as an approximate prediction band. Its statistical interpretation depends on assumptions about the errors. It is not a guarantee that a fixed percentage of any data set lies inside the band.

### 2. Interpolation with `interp1` to `interpn`

Interpolation using `interp1` to `interpn` ([YouTube](https://youtu.be/9n5acUZRl34)). These functions share one pattern: give the data points and the query points, and get the model values at the query points. `interp1` handles one input, `interp2` two inputs, `interp3` three, and `interpn` any number.

The one dimensional example connects four data points. Linear interpolation draws a straight line between neighbors. Queries outside the data range require care: depending on the method and options, they may return `NaN` or an extrapolated value. Choose the behavior explicitly. The video compares `linear`, `nearest`, `next`, `previous`, `pchip`, `cubic`, `makima` and `spline`. Available methods differ between functions and MATLAB releases. In its higher dimensional examples, the video uses `linear`, `nearest`, `cubic` and `spline`.

The two dimensional example samples the `peaks` function on an 11 by 11 grid from $-5$ to $5$. `interp2` then evaluates it on a grid with a spacing of 0.5 (441 points), and the `cubic` and `spline` methods give smoother surfaces than `linear`.

### 3. Interpolation with `interpft`

Interpolation using `interpft` ([YouTube](https://youtu.be/Rzc-8QgjawQ)). This function uses the Fourier transform. It moves the data to the frequency domain, interpolates there, and returns to the original domain with the inverse transform. This works well for data with a periodic pattern.

The example takes 30 evenly spaced samples of $f(x) = \sin(2x)\cos(x)$ up to $x = 4\pi$, and asks `interpft(y, n)` for 100 and then 300 points. `interpft` returns only the new values, so the video builds the matching $x$ values from the sample spacing and trims the output to the same length. At the end, these steps become a small helper function, `myinterpft`.

### 4. Interpolation with cubic splines

Interpolation using cubic splines ([YouTube](https://youtu.be/QdAcdStzsh4)). The `spline` function gives the same result as the `spline` method of `interp1`, and it can also return the piecewise polynomial form of the model.

The first example fits a smooth curve through a few points and shows how the curve changes as points are added. The second example builds a smooth path in the plane. A source at $(0, 0)$, a destination at $(5, 5)$ and a few random handle points are joined, and $x$ and $y$ are each interpolated as functions of a parameter $t$ from 0 to 1:

```matlab {title="Interpolate a smooth path with cubic splines"}
t  = linspace(0, 1, numel(x));
tt = linspace(0, 1, 100);
xx = spline(t, x, tt);
yy = spline(t, y, tt);
```

The video explains that this idea is useful in mobile robot path planning and optimal control. A metaheuristic such as a genetic algorithm can then search among the handle points for the best smooth path. A related example is [Optimal Robot Path Planning Using PSO in MATLAB](https://yarpiz.com/path-planning-ypap115).

### 5. Scattered data with `griddata`

Interpolation using `griddata` in 2D and 3D spaces ([YouTube](https://youtu.be/ELudRu1UZAg)). `griddata` works like `interp2` and `interp3`, but the data points do not need to lie on a grid. Its methods are `linear`, `nearest`, `natural` (natural neighbor interpolation, based on triangulation), `cubic` and `v4`.

The example draws 100 random points with $x$ and $y$ between $-3$ and $3$, computes a function value at each point and shows them with `plot3`. `griddata` then evaluates the model on a regular grid with a spacing of 0.25 (625 points), and `mesh` shows the surface. The `cubic` method gives a smooth result. The video also mentions `griddatan` for data with more input variables.

## Choosing a method

- Noisy data and a simple model: `polyfit` and `polyval`.
- Data on a grid: `interp1`, `interp2`, `interp3` or `interpn`.
- Periodic, evenly spaced data: `interpft`.
- A smooth curve or path through given points: `spline`.
- Scattered data in two or three dimensions: `griddata`.

## Check the model

Plot the original samples together with the fitted or interpolated curve. High degree polynomials and smooth splines can oscillate between samples, so a smooth result is not automatically a good model. Keep extrapolated values separate from values inside the sampled region. For noisy data, compare fits using points that were not used to estimate the coefficients.

## How to cite

Mostapha Kalami Heris, Interpolation and Curve Fitting in MATLAB: Video Tutorial (YPTNM190520S07), Yarpiz, 2019. https://yarpiz.com/interpolation-video-tutorial-yptnm190520s07