• # A Brief Introduction to Lerp

General and Gameplay Programming

Linear interpolation (sometimes called 'lerp' or 'mix') is a really handy function for creative coding, game development and generative art.

The function interpolates within the range [start..end] based on a 't' parameter, where 't' is typically within a [0..1] range.

For example, divide 'loop time' by 'loop duration' and you get a 't' value between 0.0 and 1.0.

Now you can map this 't' value to a new range, such as lerp(20, 50, t) to gradually increase a circle's radius, or lerp(20, 10, t) to gradually decrease its line thickness.

Another example: you can use linear interpolation to smoothly animate from one coordinate to another. Define a start point (x1, y1) and end point (x2, y2), then interpolate the 'x' and 'y' dimensions separately to find the computed point in between.

Or use linear interpolation to spring toward a moving target. Each frame, interpolate from the current value to the target value with a small 't' parameter, such as 0.05.

It's like saying: walk 5% toward the target each frame.

A more advanced example, but built on the same concept, is interpolating from one color (red) to another (blue).

To do this, we interpolate the (R, G, B) or (H, S, L) channels of the color individually, just like we would with a 2D or 3D coordinate.

Another quick example is to choose a random point along a line segment.

There are lots of ways to use linear interpolation, and lots more types of interpolation (cubic, bilinear, etc). These concepts also lead nicely into areas like: curves, splines and parametric equations.

Source code for each of these examples is available here: https://gist.github.com/mattdesl/3675c85a72075557dbb6b9e3e04a53d9

Matt DesLauriers is a creative coder and generative artist based in London. He combines code and emergent systems to make art for the web, print media, and physical installations.

Note:
This brief introduction to lerp was originally published as a Twitter thread and is republished here with the kind permission of the original author.

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## User Feedback

Very cool article!  I use linear interpolation to smooth out visual movement, it works very nicely with variable render rates. It's also nice for visual effects like shown above.

Thanks for sharing!

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Nice article! I loved the visual examples.

I can't say how many times lerp has been the tool of choice to achieve zillions of nice looking effects using shaders, plus is the basis of rasterization!

Keep the good work

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Note that your "springing toward" example (lerping a fraction of the remaining distance each time) is highly frame-rate dependent.  So if your game isn't frame-rate locked (i.e. web-games running on machines with different monitor refresh rates, or even a locked 60 fps game running on a machine too slow to support it) different people may see different behaviors.  The spaceship rotation in George Prosser's DRILL_BIT, for instance, is much harder to use on a slow machine.

And the way you're doing it, scaling by the deltaTime, will only partly correct for that.  For instance, if your framerate is twice as fast (deltaTime is half the length) then you actually need the square root of the rate (you lerp twice in the same time, essentially multiplying by the factor twice, i.e. squaring it).  In the general case you need Math.pow:

// Return a lerp factor for the given frame time which will result in
// the value getting convergenceFraction of the way to the target in
// smoothTime time units.  I usually just use 0.9 or 0.95 for the fraction
// and vary the smoothTime to get the effect I want.
function smoothOver(dt, smoothTime, convergenceFraction) {
return 1 - Math.pow(1 - convergenceFraction, dt / smoothTime)
}

For further adventures in non-linear interpolation things, Squirrel Eiserloh's Fast and Funky 1D Non-linear Transformations GDC talk is well worth a watch.  Although note that with his later (curved) functions, he only mumbles once that he's "normalizing" them (meaning scaling them up so they are exactly one unit tall).  Without knowing that, if you try to use some of the formulas he gives as written, you'll end up with something very flat that does almost nothing.  But he gives a good intuitive introduction to how to think about constructing some of these things.

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• Welcome to the first part of multiple effect articles about soft shadows. In recent days I've been working on area light support in my own game engine, which is critical for one of the game concepts I'd like to eventually do (if time will allow me to do so). For each area light, it is crucial to have proper soft shadows with proper penumbra. For motivation, let's have the following screenshot with 3 area lights with various sizes:

Fig. 01 - PCSS variant that allows for perfectly smooth, large-area light shadows

Let's start the article by comparison of the following 2 screenshots - one with shadows and one without:

Fig. 02 - Scene from default viewpoint lit with light without any shadows (left) and with shadows (right)

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Trivial, yet should be mentioned here.
inline float ShadowMap(Texture2D<float2> shadowMap, SamplerState shadowSamplerState, float3 coord) { return shadowMap.SampleLevel(shadowSamplerState, coord.xy, 0.0f).x < coord.z ? 0.0f : 1.0f; } Fig. 03 - code snippet for standard shadow mapping, where depth map (stored 'distance' from lights point of view) is compared against calculated 'distance' between point we're computing right now and given light position. Word 'distance' may either mean actual distance, or more likely just value on z-axis for light point of view basis.

Which is well known to everyone here, giving us basic results, that we all well know, like:

Fig. 04 - Standard Shadow Mapping

This can be simply explained with the following image:

Fig. 05 - Each rendered pixel calculates whether its 'depth' from light point is greater than what is written in 'depth' map from light point (represented as yellow dot), white lines represent computation for each pixel.

Percentage-Close-Filtering (PCF)
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Fig. 06 - Percentage close filtering (PCF) results in nice soft-edged shadows, sadly the shadow is uniformly soft everywhere

Clearly, none of the above techniques does any penumbra/umbra calculation, and therefore they're not really useful for area lights. For the sake of completeness, I'm adding basic PCF source code (for the sake of optimization, feel free to improve for your uses):
inline float ShadowMapPCF(Texture2D<float2> tex, SamplerState state, float3 projCoord, float resolution, float pixelSize, int filterSize) { float shadow = 0.0f; float2 grad = frac(projCoord.xy * resolution + 0.5f); for (int i = -filterSize; i <= filterSize; i++) { for (int j = -filterSize; j <= filterSize; j++) { float4 tmp = tex.Gather(state, projCoord.xy + float2(i, j) * float2(pixelSize, pixelSize)); tmp.x = tmp.x < projCoord.z ? 0.0f : 1.0f; tmp.y = tmp.y < projCoord.z ? 0.0f : 1.0f; tmp.z = tmp.z < projCoord.z ? 0.0f : 1.0f; tmp.w = tmp.w < projCoord.z ? 0.0f : 1.0f; shadow += lerp(lerp(tmp.w, tmp.z, grad.x), lerp(tmp.x, tmp.y, grad.x), grad.y); } } return shadow / (float)((2 * filterSize + 1) * (2 * filterSize + 1)); } Fig. 07 - PCF filtering source code

Representing this with image:

Fig. 08 - Image representing PCF, specifically a pixel with straight line and star in the end also calculates shadow in neighboring pixels (e.g. performing additional samples). The resulting shadow is then weighted sum of the results of all the samples for a given pixel.

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To understand problem in both previous techniques let's replace point light with area light in our sketch image.

Fig. 09 - Using Area light introduces penumbra and umbra. The size of penumbra is dependent on multiple factors - distance between receiver and light, distance between blocker and light and light size (shape).

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float ShadowMapPCSS(...) { float averageBlockerDistance = PCSS_BlockerDistance(...); // If there isn't any average blocker distance - it means that there is no blocker at all if (averageBlockerDistance < 1.0) { return 1.0f; } else { float penumbraSize = estimatePenumbraSize(averageBlockerDistance, ...) float shadow = ShadowPCF(..., penumbraSize); return shadow; } } Fig. 10 - Pseudo-code of PCSS shadow mapping

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// Input parameters are: // tex - Input shadow depth map // state - Sampler state for shadow depth map // projCoord - holds projection UV coordinates, and depth for receiver (~further compared against shadow depth map) // searchUV - input size for blocker search // rotationTrig - input parameter for random rotation of kernel samples inline float2 PCSS_BlockerDistance(Texture2D<float2> tex, SamplerState state, float3 projCoord, float searchUV, float2 rotationTrig) { // Perform N samples with pre-defined offset and random rotation, scale by input search size int blockers = 0; float avgBlocker = 0.0f; for (int i = 0; i < (int)PCSS_SampleCount; i++) { // Calculate sample offset (technically anything can be used here - standard NxN kernel, random samples with scale, etc.) float2 offset = PCSS_Samples[i] * searchUV; offset = PCSS_Rotate(offset, rotationTrig); // Compare given sample depth with receiver depth, if it puts receiver into shadow, this sample is a blocker float z = tex.SampleLevel(state, projCoord.xy + offset, 0.0f).x; if (z < projCoord.z) { blockers++; avgBlockerDistance += z; } } // Calculate average blocker depth avgBlocker /= blockers; // To solve cases where there are no blockers - we output 2 values - average blocker depth and no. of blockers return float2(avgBlocker, (float)blockers); } Fig. 11 - Average blocker estimation for PCSS shadow mapping

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Fig. 12 - PCSS shadow mapping in practice

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