Now, it’s time to design a custom pattern to illustrate the shader. Since all vertex and fragment is “blind” to others, we have to script a code with different manner from the concurrent programming. Thus, the position and color of a vertex should be defined with its own attributes and the shared uniform values. Here, I’ll address several techniques. For more information, please visit https://thebookofshaders.com

Basic Functions

• step(th, x): return $$1$$ if $$\rm{th} \it < x$$, otherwise $$0$$.
• smoothstep(th1, th2, x): return $$0$$ if $$x < \rm{th}_1$$, $$1$$ if $$\rm{th}_2 \it < x$$, otherwise smoothed interpolated value between [0, 1]
• mathematical operation
• abs(x): return absolute value, $$\|x\|$$
• sin, cos, tan, ...: trigonometric functions
• min(x, y): return the smaller value
• max(x, y): return the larger value
• pow(x, y): return the value of $$x$$ to the power of $$y$$; $$~x^y$$
• dot(v, w): return the dot product of vectors $$v$$ and $$w$$; $$~v \cdot w$$
• cross(v, w): return the cross product of vectors $$v$$ and $$w$$; $$~v \times w$$
• mod(x, y): return the remainder of a division $$x/y$$; $$~x - y * \rm floor(\it x/y \rm)$$
• fract(x): return the fractional part of $$x$$; $$~x - \rm floor(\it x \rm)$$
• mix(x, y, r): return the interpolated value between $$x$$ and $$y$$ with a weight $$r$$
• clamp(x, th1, th2): return the truncated value of $$x$$ between $$\rm{th}_1$$ and $$\rm{th}_2$$, i.e., $$\rm min(max( \it x, \rm{th}_1), \rm{th}_2)$$
• length(v): return the Euclidean length of vector $$v$$; $$~\|v\|$$
• distance(v, w): return the Euclidean length of vector $$v-w$$; $$~\|v-w\|$$

Using the above fundamental functions, several techniques are addressed below.

Gradient color shows a transition between multiple colors. To implement gradient color, we need $$n$$ colors and $$(n-1)$$ weight variables. mix makes you implement simple gradient easily.

varying vec2 v_position; // [0, 1]

void main() {
vec3 color1 = vec3(1.0, 0.5, 0.5);
vec3 color2 = vec3(0.5, 1.0, 1.0);
gl_FragColor = vec4(mix(color1, color2, v_position.x), 1.0);
}


Or, you can use smoothstep so that the position of gradient boundary can be tuned.

varying vec2 v_position; // [0, 1]

void main() {
vec3 color1 = vec3(1.0, 0.5, 0.5);
vec3 color2 = vec3(0.5, 1.0, 1.0);
float ratio = smoothstep(0.3, 0.7, v_position.x);
gl_FragColor = vec4(mix(color1, color2, ratio), 1.0);
}


Also, you can manually design the smoothing function.

varying vec2 v_position; // [0, 1]

float my_interp(in float a, in float b, in float x) {
float z = (x-a)/(b-a);
return z < 0.5 ? 2.0 * z * z : 1.0 - pow(-2.0 * z + 2.0, 2.0) / 2.0;
}

void main() {
vec3 color1 = vec3(1.0, 0.5, 0.5);
vec3 color2 = vec3(0.5, 1.0, 1.0);
float ratio = my_interp(0.0, 1.0, v_position.x);
gl_FragColor = vec4(mix(color1, color2, ratio), 1.0);
}


If you want to generate a gradient using three colors, use mix twice with two weight variables.

varying vec2 v_position; // [0, 1]

void main() {
vec3 color1 = vec3(1.0, 0.5, 0.5);
vec3 color2 = vec3(0.5, 1.0, 1.0);
vec3 color3 = vec3(0.2, 0.5, 1.0);
float ratio1 = smoothstep(0.0, 0.6, v_position.x);
float ratio2 = smoothstep(0.4, 1.0, v_position.x);
gl_FragColor = vec4(mix(mix(color1, color2, ratio1), color3, ratio2), 1.0);
}


Repetitive Pattern

When zooming out on most complicated natural texture, you can see they are patterned. To divide a range [0, 1] into multiple [0, 1] s, we use fract() function.

varying vec2 v_position; // [0, 1]

void main() {
float ncol = 3.0;
float nrow = 2.0;
float x1 = fract(ncol * v_position.x);
float y1 = fract(nrow * v_position.y);
float idx_x1 = floor(ncol * v_position.x);
float idx_y1 = floor(nrow * v_position.y);

gl_FragColor = vec4(x1, y1, (idx_x1 + idx_y1) / 5.0, 1.0);
}


Likewise, if you want to generate a pattern inside a pattern, divide grids twice.

varying vec2 v_position; // [0, 1]

vec4 divide_cell(in vec2 parent, in vec2 size) {
vec4 child = vec4(0.0);
child.xy = fract(size * parent); // x, y position
child.zw = floor(size * parent); // x, y index
return child;
}

void main() {
vec4 xy1 = divide_cell(v_position, vec2(3.0, 2.0));
vec4 xy2 = divide_cell(xy1.xy, vec2(2.0, 2.0));

gl_FragColor = vec4(xy2.x, xy2.y, mod(xy2.z + xy2.w, 2.0), 1.0);
}


Random

Unfortunately, GLSL does not provide a true random number generator. Instead, we can generate a pseudo random number with a sinusoidal function of high amplitude with fract. fract transforms the value into the range [0, 1], and the sinusoidal function produces randomness by generating a different slope for each [0, 1] piece. Also, fract(x^2) can generate random number since their slopes are different for each piece. However, unlike the sinusoidal function, which is bounded on [-1, 1], x^2 can exceed the maximum value of Float, thus outputting incorrect value.

float random(in float x) {
return fract(sin(x)*100000.0);
}


Using MATLAB, I’ve generated the histogram of the above random variable.

x = linspace(0, pi, 10000);

figure();
set(gcf, 'Position', [680 557 720 480])

z = sin(x)*100000;
%% z = x.^2*100000;
%% z = x*100000;
X = z - floor(z);

subplot(211);
histogram(X, 'Normalization', 'pdf');
xlim([0, 1]);
grid on
xlabel('Value, X');
ylabel('PDF, f(X)');
set(gcf, 'color', 'w');

subplot(212);
plot(x, X, '.');
xlim([0, pi]);
ylabel('Value, X');

formula result
$$\rm sin(\it x)$$
$$x^2$$
$$x$$

Above, the histogram of $$\rm sin(\it x \rm)$$, $$x^2$$, $$x$$ are shown. As mentioned before, sinusoidal and power functions show randomness, whereas linear function has a pattern. Please notice that they have a uniform distribution.

To generate a random number from two or more dimensional vector, we use dot product to produce a scalar number.

float random(in vec2 x) {
return fract(sin(dot(x, vec2(12.9898,78.233)))*43758.5453123);
}


The above magic numbers can be chosen manually so that randomness is shown well.

Noise

If we called a single independent noise as random, noise means the interpolated value between random values. Thus, noise functions have a continuity, whereas random functions show discontinuity. Below are the example of random and noise.

random noise

As mentioned above, all vertex and fragment is blind to others, so each fragment can not read the colors nearby. However, since the random function of GLSL generates pseudo-random output rather than true random, thus each fragment can read the value of adjacent fragments.

float random (in vec2 x) {
return fract(sin(dot(x, vec2(12.9898,78.233))) * 43758.5453123);
}

float noise (in vec2 x) {
vec2 i = floor(x);
vec2 f = fract(x);

// Four corners in 2D of a tile
float a = random(i);
float b = random(i + vec2(1.0, 0.0));
float c = random(i + vec2(0.0, 1.0));
float d = random(i + vec2(1.0, 1.0));

// Cubic Hermine Curve. Same as SmoothStep()
vec2 u = smoothstep(0., 1., f);

// Mix 4 coorners percentages
return mix(mix(a, b, u.x), mix(c, d, u.x), u.y);
}


In real world, a signal is the sum of sinusoids of multiple frequencies, and we can compute the amplitude of sinusoids of the specific frequency by Fourier transform. In noise pattern, a division number corresponds to a frequency of Fourier transform. Therefore, when we add multiple noises of different division size, a natural texture can be imitated.

float random (in vec2 x) {
return fract(sin(dot(x, vec2(12.9898,54.233))) * 43758.5453123);
}

float noise (in vec2 x) {
vec2 i = floor(x);
vec2 f = fract(x);

float tl = random(i); // top-left corner
float tr = random(i + vec2(1.0, 0.0)); // top-right corner
float bl = random(i + vec2(0.0, 1.0)); // bottom-left corner
float br = random(i + vec2(1.0, 1.0)); // bottom-right corner

vec2 u = smoothstep(0., 1., f);

// Mix 4 coorners percentages
return mix(mix(tl, tr, u.x), mix(bl, br, u.x), u.y);
}

#define LEVELS 6
float fractal_noise (in vec2 x, in float scaling_amp, in float scaling_freq) {
float n = 0.;
float amplitude = 1. - scaling_amp; // ensure that the maximum value is 1.

for (int i = 0; i < LEVELS; ++i) {
n += amplitude * noise(x);
amplitude *= scaling_amp;
x *= scaling_freq;
}

return n;
}

void main() {
gl_FragColor = vec4(vec3(fractal_noise(2.*v_position, 0.5, 2.)), 1.);
}


Also, you can compose multiple fractal noises and time variable to illustrate flowing textures such as smoke.

void main() {
vec3 color = vec3(0.5, 0.6, 0.7);

float n = fractal_noise(2.*v_position, 0.5, 2.);
float m = fractal_noise(2.*v_position + vec2(0.1, 0.12)*u_time + n, 0.5, 2.);

gl_FragColor = vec4(mix(vec3(0.), color, m), 1.);
}


Tags:

Categories:

Updated: