Painting a Landscape with Maths

hi welcome back certainly glad you could join me today my name is inigo quiles and i thought that today we would do a fantastic painting that it's a lot of fun not by using traditional paint or brushes or digital ones but by using mathematics instead that's right indeed we will define colors and shapes through formulas that when composed together into one large formula will result in this mathematical painting that you see so i told you what let me run all the formulas across the screen that you will need to do this painting along with me today and while i do that let me assure you that painting with maths is not difficult if done one step at a time as we are going to do today and it's most definitely lots of fun so let's erase all this mathematical paint to get the canvas ready to go so we can start from the beginning plotting a 2d surface f that is just some linear function of x and c the two horizontal coordinates of space as you see i'm using y for the app direction today and i hope i don't annoy the viewers that are in the c up team too much now such a linear function is a polynomial of degree 1 but if we progressively increase the degree of the polynomial we can provide our surface with more and more undulations this is great as a first sketch for the terrain but to paint a complex terrain with thousands of hills and tiny bumps this polynomial would need thousands of terms of increasing degree which we need to avoid because that's expensive to evaluate and difficult for the computer to plot so instead we are going to make the terrain a piecewise function such that we will assign a unique but small and manageable polynomial to each one by one square tile of the domain [Music] i'm only showing four such styles here but the domain of the function is infinite of course now in order to ensure that all these polynomial tiles connect nicely to each other we will construct them in this form which is a polynomial of degree 3 in 4 parameters a b c and d where each parameter corresponds directly to the function's height at each one of the four corners of each tile so by sharing the coefficients at the vertices of the tiles we can ensure the connectivity of all the polynomials [Music] we make these coefficients look random by taking the integer coordinates of each vertex i and j dividing them by some irrational number such as pi computing the fractional part and that's what the curly braces mean here multiplying them by 50 and then combining them together and computing their fractional part yet again [Music] now using this particular expression is not critical and you can replace it by any function that shuffles i and j sufficiently and produces a random enough locking value for each vertex [Music] on the other hand these polynomial components here are intentional and force the derivatives of the surface along the connecting edges to be zero so they always match across neighboring tiles without them we would have a continuous surface but not a smooth one and speaking of tangents let me degrease a little bit here to talk to you about this s shape that runs from 0 to 1 because when we generalize it to run between any arbitrary points a and b we call it smooth step and becomes one of the most useful mathematical brushes that we have you will see today applied multiple times in the form of capital s of a b x and since i'm talking about useful mathematical brushes functions like f that undulate both smoothly and randomly are usually called noise and are very important in the making of visual effects for films which are often based on heavily art directed mathematics and so noise shows up everywhere from the creation of clouds and smoke to vegetation and mountains surface detail and dirt and almost anything so let's give it a proper name in this video and refer to it with capital n from now on so we can use f for the function of the terrain itself that we are designing which we are going to continue to work on and add some detail to make it more like an actual terrain so let's pack the x and z coordinates of space as a 2d vector p sub x c and then compress another noise function horizontally by multiplying p by 2 and then squashing it vertically by a factor of 2 as well to keep its features undistorted next we rotate it so the two functions domains do not align to each other the angle of rotation is arbitrary and not important so i will just use the pythagorean triple 345 to define one i often use this triple or 8 15 17 or some other of the small pythagorean triples as a way to quickly fill up rotation matrices without resorting to sines and cosines [Music] all right the last step here is to add this scale and rotated polynomial back to the original one to create new undulations on it this is why preventing the alignment of the two functions was important and so we will repeat this a few times with larger and larger compression squashing and rotation factors in order to get smaller and smaller undulations which provide finer and finer detail for our surface and yes if this process of summing undulating waves of different frequencies made you think of a fourier series or a discrete cosine transform then you are definitely having a good intuition here and you should check a link i left in the description of the video to an article where i do a frequency analysis of this summation and explains why it does resemble that terrain [Music] in fact let's start now evaluating the function in its whole domain and scale this domain by 2000 and the range of the function by 600 so from now on one unit of space equals about 1 meter so we can sculpt the terrain in a more intuitive way also i will reserve the uppercase f for the canonical non-scaled fractal sound because we are going to be using it so many more times today okay with that in place let's now change gears and paint some lighting starting with our strongest light or the key light the sun [Music] we will use the derivatives of f as a way to determine which areas of the surface are facing towards or away from the sun concretely our surface f will have a derivative with respect to x which gives us the slope and tangent in the x direction and the derivative with respect to z which gives us a slope and a tangent in the z direction now the cross product of these two tangents is perpendicular to the surface so let's call it the normal vector and note that it is exactly the direction of this vector that we should compare to the sun direction here in yellow with a dot product in order to determine whether the sun is illuminating the surface or not concretely on the left side of this hill both are pointing in the same general direction so the dot product is close to 1 which means that this must be the sunny side of the hill but as we move towards the right side of the hill the two vectors agreement and hence their dot product decreases until it becomes negative indicating that this portion of the heel is in the shade so if we take the dot product and when positive multiplied by brown then we will get a color within a palette of shades of brown that we can assign to each point on the surface and finally start getting some sense of volume for the terrain and while this is a great start the image is still too busy and difficult to read i think but painting some shadows will help with that so let's define a function sh that returns 0 when there is an intersection between the line of sight to the sun that starts at p and our polynomial surface and that returns 1 when there isn't any such intersection then when we multiply our surface color with it we can finally see shadows and with them the larger the structures of our terrain as we wanted making it easier to understand its overall shape but let's now replace sh of p by something better something that creates a transition zone between 0 and 1 instead we do that by constructing a new function r of t that is equal to the sign's distance from any point in the terrain f to a point in the line t units away from p all divided by t the minimum of r smooth stepped to the range 0 to 1 will be a pretty good estimation of the penumbras cast by the terrain see how we get now beautiful soft shadows very nice however like most fractal structures the physical scale of this terrain is currently a bit ambiguous without any trees or humans for reference even after rotating and scaling the domain of our terrain function to get a closer look at it viewers still can't possibly know if these are large hills that are very far or tiny bumps that we are very closely looking at to fix this we are going to compute a decaying exponential of the distance to any point on the surface t and use it to mix in some gray into our surface color so that we gradually reduce the contrast and saturation in the most distant parts of the painting which is what happens when you look at a real landscape and indeed we finally do have a sense of scale for the terrain don't we now of course we can and should play with the attenuation factor of our exponential brush and decide how hazy or clear we want our landscape to be today all options are open and will look beautiful and actually that's no coincidence exponentials do model the transmittance of light in the real atmosphere very well and while our paintings don't need to be faithful to reality sometimes it's just convenient to learn from nature itself alright something fun now let's replace this exponential with three different exponentials one footage of our red green and blue color channels that way by adjusting the three damping constants independently we just gave the atmosphere a beautiful blue tint in the medial range distances before it becomes gray or monochrome here at the back great now that we have the terrain math done let's lock the composition of the piece by choosing a good section of this infinite terrain to frame into after scouting for a bit by rotating and translating the surface i found this location with only two depth layers with this hill popping out as a foreground element well separated from the background which i think makes it very photogenic but the choice of location is totally up to you choose one that you like and don't fall for advice about mystical spirals and perfect compositions that only work by moving and stretching the spiral after the fact or most irrationally sacred golden ratios supposedly linked to the beauty of the universe or whatever all i'm saying is just go with what feels right to you for example even though i like the location i think i want something a bit more dramatic so let's sculpt a cliff in it by using the function c which takes everything above 500 meters with a smooth step and then pulls it up by some extra 90 meters but of course doing so changes the derivatives of the terrain which in turn we needed for lighting so we will compute the derivatives of the new surface by using the chain rule which involves the derivative of the smooth step scoped operation and the derivatives of the terrain itself before we applied it okey dokey so now that we have our framing locked we can continue working on the painting's composition which means we have to make more decisions and the most important decision of all is where the sun is in the sky and what we want the lighting to look like so first if we describe the sun's direction or position in the sky s in spherical coordinates let's push theta close to pi halves so the sun is low and gives us long dramatic shadows from there let's now explore how different values of phi affect the composition for example at this very moment the light comes from the left and a bit to the front but if we rotate it to the right side then we have a different set of shadows although in this case we lose the nice separation that we had between the front heel and the background clips if on the other hand we rotate the light to the back we split the painting in a dark and a bright area which i don't like personally so i settled on a sand coming from the back left side this illuminates both parts of the foreground and the background but still keeps them separated thanks to the atmospheric coloring that we painted it also creates these very cool silhouettes in the front heels here and in the background clips here but again this is all up to you this is your little mathematical world and you can do anything you want in it so just have fun and so next we will paint a pretty sky for our world we start by making the background of the canvas blue and then we use a linear function of the vertical component of the direction of each point in the canvas v so that the upper part of the sky gets darker and more saturated than those closer to the horizon then we are going to place a horizontal plane at about two and a half kilometers in altitude and evaluate the same polynomial function that we used for the terrain surface on it but with a different scale then we remap its negative 1 to positive 1 range into 0 to 1 with a smooth step and display it as shades of gray [Music] now we use the value of the polynomial to drive a weighted color average where small values of the polynomial map to blue and large values map to white which creates some happy little clouds in the sky pretty cool although a sky this hazy would never produce the strong light and shadows that we painted in the terrain so let's clear up the sky a bit just like so which is very easy since we can control both how large the clouds are and also how fuzzy or sharp their edges are by changing the parameters of the smooth step and as you can see we keep coming back to our friend the smooth step function [Music] but anyway let's now soften these clouds by reducing their contribution to 40 in the color mixing formula and lastly we scale the domain of the function by 2 so the clouds become twice smaller just like that and since we know that we will be painting more layers of cloud on top i don't think we need to add more details to this one so let's stop there move on and quickly adjust the overall contrast and saturation of the painting before adding any more lighting and details that's because i feel the image is a bit too flat right now so i'm going to make it more vibrant and contrasty by using well wait for it smooth step again the smoothest step makes the dark colors darker and the bright ones brighter pretty much like you would do when adjusting color curves in a photo editing program [Music] very cool now let's take care of these large patches of flat color here because even though these areas can't be reached by the sun they shouldn't be in complete darkness but partially illuminated by the sky's blue light an effect that should be strongest when the surfaces are facing directly to the sky so let's capture that idea and the light by taking the normal's vertical component scaling and biasing it by a half and then multiplying it by the sky's blue color and finally adding it to our lighting formula so much better indeed by the way the 10 factor here in the formula is because over time i realized that this ratio of sunlight to skylight produces realistic lighting now while we finally see some detail in the shadows i still think we don't have enough definition especially in the cliffs here so let's brighten them up even more by considering not only the light coming from the sun but also that coming from the nearby areas of the terrain that are strongly illuminated by the sun in that bouncing effect the sunlight not only has changed to approximately the opposite direction let's call it b but also has gotten multiplied or tinted if you want by the brown color of the terrain itself so there we are see again the difference between before and after applying this warm bounce light coloring very nice all right with the composition locked now is when we start making this painting pretty so i'll tell you what let's split the color formula between material and lighting colors for convenience and now add some basic grass layer to the material by mixing some green into the current brown color that we have with our usual color mixing formula that is driven by the vertical component of the terrains normal composed with a smooth step in order to isolate flat areas where the grass can grow it is that easy and of course we can and should experiment with changing the parameters of the smooth step to control how much grass coverage we want but today let's actually not spend too much time tweaking that because once we cover most of the terrain with trees this layer of green will be there only as a vague indication of a grassy ground but before we paint trees i think i wouldn't actually expect trees to grow in such a rocky terrain like the one we have here so what about we filter out some of these undulations say those in the range of 64 to half a meter which is approximately the scale at which vegetation happens and because the series that we used to construct the surface f doubled frequency with each term all we need to do really is just omit the right seven terms that span that 64 to half a meter wavelength range [Music] there we go the effect is similar to band pass filtering the terrain signal but we are doing it at synthesis time which is easier than doing it after the fact anyway now we are finally ready to plant our trees so let's start by dividing the x and z coordinates of space by 2 and rounding the result down to the closest integer this tiles space into squares of 2 meters where m is the 2 dimensional index of each tile with it we can compute the center of the tile and with the terrain surface evaluated at that center we can create a unique local coordinate system for each style w and finally in this coordinate system we can evaluate a sphere's sign distance function with a radius close to one [Music] now let me talk about what just happened here because even though we just painted millions of spheres in the canvas unlike traditional painting or even unlike traditional computer graphics where each tree needs to be painted one at a time here we are only evaluating one sphere as part of our large mathematical painting but because of the tiling of the coordinates this one sphere function has become periodic and covers the whole space creating infinite detail but while that's fantastic nature is rarely as perfectly periodic and regular as what we have here now so let's add some variation similarly to how we randomized our terrain polynomial we take the two-dimensional index of each style make it one-dimensional take the fractional part of their division by two irrational numbers multiply that by itself and take the fractional parts again this is clearly a bit of an arbitrary function again so feel free to design your own all we need is something that feels random because with it we are going to shift the center of the sphere by plus minus half meter within its 2x2 tile in order to make this field of trees look less regular great now let's stretch the spheres vertically into ellipsoids so we get something that looks less like big bushes and more like trees [Music] this seems simple but there is an interesting technical catch here the implicit equation of an ellipsoid that is most natural is not a distance field making it very inconvenient for rendering on the other hand the actual sdf of an ellipsoid requires solving a degree 6 equation which has no closed form in radicals and numerical solvers have problems with but the sdf of symmetric ellipsoids like the ones we probably need today can however be solved analytically with aquatic equation but that turns out to be too slow for us so in this painting i'm using this other approximation that i developed that is completely robust and efficient enough in any case let's now move the center of the ellipsoids up so their lowest part sits on the ground so we get to see more of a tree shape however this introduces some problems here in the cliffs because all the ellipsoids that were hidden under them are now poking out can you see them that feels wrong but we won't erase them yet and let them be instead we can take care of them later when we have finalized painting the trees and instead let's go fix another thing that is really bothering me which is that the ground here between the trees feels too bright given that in such a dense forest i would expect very little of the ambient light to make it down there so let's bring up the color formula again and compute a smooth step of the altitude to isolate the valleys from the cliffs and multiply it by the ambient light to reduce it by 80 percent [Music] and in fact for the same reason we will also modify the lighting formula of the trees but with a linear function that goes from zero at the bottom of the ellipsoid to one at its center with it we also indicate that not much of the light makes it to the lowest parts of the forest and yeah indeed this looks so much better now so now let's work on giving this forest a more organic feel to it as a first step we will change the height and the width of each tree by using the same pseudo-random function that we designed earlier but with a different offset so we decorrelate the random sizes from the random three positions indeed different offsets produce different sets of random values but they are all equally good as you can see so it doesn't matter much which values we pick however once we have chosen some it can be crucial to commit to them i learned this the very hard way when i was working at pixar generating mathematical art for the movies so the story is that i based the look of the vegetation of the film that we were making on the randomness generated by such an arbitrary number like we just did which i made equal to my phone number because why not now that was all fine and dandy until the day i changed to a new carrier and phone number with it and i felt the urge to update the formula accordingly this should have not been any problem since the new randomness generated from it was as valid of a randomness as the old one but well the truth is everyone freaked out when tens of film shots that have been approved already for the big screen suddenly looked all different overnight it was crisis time and while in the end we did actually and unexpectedly keep the new phone number the moral of the story is still to always be extremely careful if you play with arbitrary formulas and numbers in a professional context anyways back to our painting and to finish sculpting the trees we will now use a noise function like the one we designed for the terrain surface but in 3d and we will define it over the whole space p rather than just in the local coordinate of each tree w then we square it to make it a bit more spiky and we add it back to the ellipsoid function this distorts its silhouette into something more organic and no longer perfectly smooth and geometrical and with the right scaling for it it will make all these ellipsoids look like happy little trees just like that actually the fact that we don't need to recreate botanically accurate trees but only get something that looks like trees is really important here please don't miss the forest for the trees and don't fall into the rabbit hole of painting with super detailed mathematical or procedural branch systems or anything like that all we need really is just the illusion of trees but because they cover the whole terrain like a carpet we no longer get to make out the shape of the terrain and the hills and the valleys in it so let's improve this by tweaking the lighting on the trees which is currently happening through their sdfs gradient or surface normal and subtee the trick is to linearly combine it with the normal of the terrain on which the tree grows and sub f say in a two to one ratio this makes the normals of the trees partially reflect the orientation of the terrain and now because the dot product at the core of our lighting equations is a linear operator linearly combining the normals of the tree and the terrain in this way is equivalent to linearly combining the terrains and trees individual lighting in the same two to one ratio or in other words the lighting on the trees now will partially look like that of the underlying terrain revealing its shape checking it how much better it looks now and while we are at it let's also take the dot product of the normal and the view direction to select the three silhouettes raise it to the fifth power to make the selection sharper multiply it by light yellow and by the vertical ramp lambda that we created earlier so the selection fades at the bottom of the trees and finally we add it to the tree color to create these beautiful highlights the cool thing is that because we are letting some of the terrain's surface normal leak into the trees these highlights are also capturing some of the beautiful silhouettes of our hills giving us a better understanding of the different layers of depth that we have in the painting lovely this is going really great but now it is time for us to make some big decisions indeed when i was at this point in the painting i realized that i had to pick a season for it i quickly sketched some winter colors which looked beautiful and also some fall colors which were also very nice but that's when i thought that these ellipsoidal trees really feel like the trees you would find in a rainforest more than alpine trees and because i didn't want to paint new trees over i decided to stick to vibrant rainforest greens and move on but once it's decided that we go with greens it's also clear that it can't all be just the same green everywhere nature uses rich color palettes so let's do the same and add some color variation to our forest we start by generating a random number again by dividing the index of each tree in the grid by pi and shuffling its fractional parts a bit then we compose it with a smooth step and use it in our usual linear color combination formula such that we give some of our happy little trees a yellow tint now let's keep adding color richness and invoke the fractal noise function that we created earlier but limit it to four terms so it is smooth then we tune its sharpness with a smooth step and let it drive the mix of the base green color with some ochre and let me put a magnifying glass here so we can see this change better [Music] beautiful lastly we will use the same fractal noise but scale differently make it almost binary with a very sharp smooth step and then use it to make some of the trees dark green and also taller and thinner indicating that there is a second species of tree here competing for space great but what i'm noticing now is that here in the background these patches of dark trees became too distracting so let's smoothly undo the effects of the last brush lambda 3 by multiplying it by 1 minus a smooth step of the distance to each stream that looks much better now doesn't it okay now we have to switch gears and do some computer graphics but bear with me it will be short and definitely worth it so the thing is we need to add some highlights to our painting so it feels more realistic because all real surfaces behave a bit like a mirror and reflect the sunlight back to us when the orientation of the surface the sun and with the viewers are correctly aligned relative to each other for example when the incoming sunlight hits our cliffs it will bounce along its normal in the reflected direction r which we can compute with a dot product and some geometry we can then measure how much of that light is actually coming straight to us the viewer's v with another dot product informing us of how much white paint we should add to the canvas we also raised it to the 9th power so that the white fades rapidly away when the alignment is not perfect then we make all these highlights more intense when the sunlight is mirrored at a shallow angle much like the road can blind you when you are driving and the sun is slow in front of you we do this with another dot product and the square root raised to the fifth power and letting as much as 95 percent of the highlight intensity depend on it now these formulas and values are not arbitrary but common practice in contemporary computer graphics because they are really great approximations to how real light and materials behave in nature and while following them is definitely not mandatory for making beautiful images i wanted to give you a bit of a sneak peek to the kind of math you will be programming if you were a rendering engineer working on films or computer games but also look indeed and these gorgeous highlights that we just got for our trees and cliffs and speaking of cliffs let's go revisit those trees that were under the cliffs that were creating some ugly and distracting patterns earlier because it seems to me now that after we randomized their sizes and positions everything looks now pretty organic to me and pretty much like vegetation so why not pretend these are some little bushes that live up there in the cliffs and welcome them to our painting because we don't make mistakes here we just have happy accidents all right let's work now on fixing and finalizing the overall look of the painting for example i feel the vegetation above the cliffs is too bright so let's select the highest elevation points in the terrain with a smooth step of y and then multiply the color of our trees by one minus half of that so that we darken them by 50 percent [Music] i also feel i want to adjust the lighting of the foreground trees and separate a bit the parts under the sun from those in the shadow so we will take a smooth step of the distance to each point in the painting and subtract it from 2 and multiply it by the sun's contribution see the difference now these were very localized tweaks but now let's do some global color adjustment like taking the painting as a whole and raising its red green and blue color channels to the powers of 1 0.9 and 1. or in other words we leave the red and the blue channels of our painting unchanged but we pull the middle greens up which makes our vegetation feel a bit more translucent and lush we also add a little bit of blue to create a sense of a thicker atmosphere which also desaturates and makes the image feel a bit more natural although this is totally up to you and your own taste speaking of taste i often like adding a bit of a photographic feel to the painting like this we consider the vector that points towards each element of the canvas and compute its dot product with the sun's direction to get a sense of the proximity of the sun to which point on the canvas if the dot product is positive we raise it to the fourth power and multiply it by orange over four before adding it to the canvas color this simulates the sangler that often happens in real photographs and film although i'm keeping it soft in intensity and hence the division by 4 because i don't want to over do it and grab too much of the viewer's attention and on the topic of attention let's guide the viewer's eyes a bit towards the center of the image like this we multiply the painting by a parabola that has its zeros at the edges of the picture and reaches a maximum of 1 at the center then we multiply it by a similar parabola that runs vertically across the screen then we take its 20th root so we flatten it towards one and finally we scale and offset it by a half darkening the borders of the painting as we wanted here's the difference before and after applying this effect and now to finish the landscape let's paint some new and more realistic clouds on top of the ones that we already have or rather than paint let's sculpt them so we convey volume more easily so let's start by building a solid block of clouds through a thin but infinitely wide box at 900 meters of altitude then we add three-dimensional noise to it which will open big holes in the cloud volume just like we did open gaps in the three ellipsoids the cloud colors right now are a direct black to white mapping of the dot product between the sunlight direction and the gradient of the cloud density and yes we will improve this in a moment but regardless the clouds won't feel well integrated with the rest of the scene until we compose them with the same atmospheric coloring and the three color exponentials that we used for the terrain and the trees also before we continue sculpting we need to make sure that the clouds feel fluffy we do this by changing the plotting algorithm which currently finds the zero level set or isosurface of our cloud density function and then applies our coloring and lighting formulas to it so instead we will accumulate cloud density and color along all the level sets as our line of sight crosses them and this is what that looks like beautiful isn't it but still mandelbrot noticed that clouds are not spheres and that their silhouettes are not smooth but fractal and he was right that's exactly why we are now going to increase the realism of these adorable clouds through the same techniques that we used for the terrain where we add multiple layers of noise of increasingly higher frequencies and lower amplitudes beautiful however as i painted this on the canvas i've applied a trick that i need to disclose so let me explain this is what our fractal clouds look like without the trick when we use their density gradient straight for illumination purposes as we have done for trees and terrain indeed clouds in reality look much softer than this because the light inside of them bounces around and scatters multiple times smoothing out all the features and leaving only the overall shape of the cloud for us to see but we won't simulate any of that and instead the trick will be to kind of low pass filter the cloud density gradient we do this by only using four components of the fractal noise when computing the gradient instead of using all the components that we are using when we compute the shape this softens the gradient and with it the lighting and makes the clouds look so much better indeed and i also think they look better when we change the scale of their domain to make them twice as large and with that let's go and work now on the lighting first we increase the contrast by changing the scale and the bias in the dot product then for every point in the cloud we will evaluate the density field at a second point a few units away towards the sun in search of negative density values that indicate the presence of a cloud at that location that could be obscure in the sun we capture such negative densities with a smooth step which also helps us control the sharpness of the transition between light and shadows and then we multiply it with the sun's contribution and look at that it's starting to look really sweet great it's time to think about what other light sources besides the sun could be illuminating our clouds and the sky comes to mind right away naturally so let's take the vertical component of the normalized cloud gradient map it linearly to the 0 to 1 range multiply it by blue and add it to our cloud color because we want the up facing areas of the clouds to be most exposed to the skylight we will also tint the bottom of the clouds in green since the trees in the terrain should reflect some green light upwards so we can just reuse the same formula and modify it for downwards point ingredients and for green color lastly we multiply the color of the clouds by a linearly decreasing function of the density so we darken their interior and i think we should also add some glare to the clouds that are closest to the sun so let's use again a dot product raise it to a big power and modulate the amount of orange that we add to the cloud color [Music] and because i think that these happy clouds look beautiful already i think we should move on and finish off the painting with some cloud shadows on the terrain now honestly we could just paint some dark arbitrary blobs over the terrain and that would definitely work but instead this time we are going to let the mathematics that we already developed do the painting for us so first we construct the line going from each point p in the landscape towards the sun and we intersect it with a horizontal plane placed at the same height as the clouds then at that intersection point q we compute the cloud density function and pass it through a smooth step to isolate negative values which we can then multiply with the sand coloring function of the trees and rocks in order to catch the shadows on them of course we can control the size and the softness of the penumbras with the parameters of the smooth step and i personally like the painting best with only a gentle shadow effect so i will leave the parameters there now because these shadows are planar projections of the actual cloud density function if we shift its domain either horizontally to displace the clouds or vertically to change their shape then the shadows on the terrain will follow and do the right thing automatically which is super cool [Music] and with that we finish the painting and while clearly this isn't the most masterful landscape painting that you have ever seen i find the fact that we painted it all with maths quite beautiful in itself if you ask me remember that all individual color lighting and shape formulas that we designed are actually composed into one large and singular function that is responsible for returning a red green and blue color triple for any x y point in the canvas actually besides the painting itself all the explanatory diagrams and plots and animations that you have seen in the video are also painted with masks and parts of the painting's formula making it a mathematical video more than a mathematical painting perhaps except for my face and the text overlays of course also there is a whole other layer of mathematics going on in order to get the computer to plot all these formulas into the screen which has to do with computer graphics so we won't touch on that today but all of this to say that it took me quite some work to make this video so thank you so much to all the fine people who support me on patreon and since you made it this far thanks to you two for sticking around until the end and spending this time with me i really appreciate it and finally i'd like to wish you happy painting with math and be well my friend

Channel: Inigo Quilez

Views: 696,291

Keywords: raymarching, mathematics, art, painting, procedural, glsl, calculus, trigonometry, cg, shader, realtime, quilez, demoscene

Id: BFld4EBO2RE

Length: 42min 0sec (2520 seconds)

Published: Sun Apr 10 2022