the day you finally use algebra: a 3d math primer

The Day You Finally Use Algebra!

Janie Clayton-Hasz

About Me

But math is hard! (Let’s go shopping!)

Math is hard. But math is fun too.

Normalized Coordinate

Systems

Cartesian Coordinates

320

480

320

480 or 568

(0,0) (1,0)

(0,1) (1,1)

self.size.width

self. size.

height

- Colors, like the screen dimensions, are based on percentages rather than absolute values.

- If you come from a graphic design background, you need to convert your 255 scale to percentages.

Algorithm Rosetta Stone

Rosetta Stone- Had the same text in

Greek, demotic, and hieroglyphics. Was used to translate hieroglyphics

- Going to do similar thing, but with math algorithms, plain English, and code

√-1 2ˆ3 ∑ π

∑5

i = 1

4i

Algoritm

I have a starting value of one. I have an end value of five. I want to multiply each value by four and add them together.

Plain English

I have a starting value of one. I have an end value of five. I want to multiply each value

by four and add them together.

var x = 0 !

for index in 1…5 { !

x += (4 * index) !

}

Code

It’s All Greek to Me

π

∆

i

θ

Constant: 3.14159…

Change between two values

Square root of negative one

Variable representing an angle

Consolidate slide, get rid of alpha and beta

√-1 2ˆ3 ∑ π…and it was delicious!

i 8 sum pi…and it was delicious!

Trigonometry

Triangles

A shape with three sides where the angles add up to 180 degrees

Everything in our world comes back to triangles

The most stable shape

Foundation of 3D graphics

Right Triangles

Pythagorean Theorem

aˆ2 + bˆ2 = cˆ2

Distance - Need to figure out better examples/stories

Triangle Formulas

Tangent

Sin

Cosine

Arctangent

Arcsin

Arccosine

Triangle Formulas

Circle Formulas

Circumference: 2πr

Area: πrˆ2

So What Can We Do Knowing This?

Change the direction a character is moving in

Check to see if the user is hitting a target area on the screen

Draw shapes and filters in specific configurations

Linear Algebra

Google “Better Explained Rotation” Matrices Complex numbers Show how to use linear algebra to do more efficient affine transforms rotation through matrix multiplications

What is Linear Algebra?

Linear Algebra allows you to perform an action on many values at the same time.

This action must be consistent across all values, such as multiplying every value by two.

What is Linear Algebra?

Values are placed in an object called a matrix and the actions performed on the values are called transforms

Linear algebra is optimized for parallel mathematical operations.

Data Types

vec2, vec3, vec4: 2D, 3D, and 4D floating point vector objects.

vec2: (x, y)

vec3: (x, y, z)

vec4: (r, g, b, a)

Data Types

mat2, mat3, mat4: 2, 3, and 4 element matrices.

mat2: Holds a 2 X 2 number matrix

mat3: Holds a 3 X 3 number matrix, used for 2D linear algebra

mat4: Holds a 4 X 4 number matrix, used for 3D linear algebra

Enter the Matrix

1.0 1.0 1.0 0 1.0 1.0 1.0 0 1.0 1.0 1.0 0 1.0 1.0 1.0 0

Column

Row

mat4 genericMatrix = mat4( !

1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 0, 0, 0, 0

);

Column

Row

Column major vs Row major??

vec4 firstColumn = vec4(1.0, 1.0, 1.0, 1.0);

vec4 secondColumn = vec4(1.0, 1.0, 1.0, 1.0);

vec4 thirdColumn = vec4(1.0, 1.0, 1.0, 1.0);

vec4 fouthColumn = vec4(0, 0, 0, 0);

mat4 myMatrix = mat4( firstColumn, SecondColumn, thirdColumn, FourthColumn

);

CGAffineTransform

Affine, Wha?? :(A transform is any function that alters the size, position, or rotation of an object on your screen.

Four types: Identity, Translate, Rotation, and Scale.

For a transform to be affine, the lines in your shape must be parallel.

CGAffine Transform Methods

CGAffineTransformMakeRotation (GLFloat angle);

CGAffineTransformMakeScale (CGFLoat sx, CGFloat sy);

CGAffineTransformMakeTranslation (CGFloat tx, CGFloat ty);

Affine Transform Rotate Talk about chaining transforms CAAffineTransform

struct CAAffineTransform { CGFloat a; GLFloat b; CGFloat c; CGFloat d; CGFloat tx; CGFloat ty

}

new point x = a * x + c * y + tx;

new point y = b * x + d * y + ty;

Show the matrix of the vector and the actual math around it

How Does This Work?

For each point in your shape, the computer uses this calculation to figure out where the point should be.

If you have a rectangle, this gets run four times: One for each point in your shape.

Refraction Fragment Shader Example

void main() { highp vec2 textureCoordinateToUse = vec2(textureCoordinate.x, (textureCoordinate.y * aspectRatio + 0.5 - 0.5 * aspectRatio)); highp float distanceFromCenter = distance(center, textureCoordinateToUse); lowp float checkForPresenceWithinSphere = step(distanceFromCenter, radius); distanceFromCenter = distanceFromCenter / radius; highp float normalizedDepth = radius * sqrt(1.0 - distanceFromCenter * distanceFromCenter); highp vec3 sphereNormal = normalize(vec3(textureCoordinateToUse - center, normalizedDepth)); highp vec3 refractedVector = 2.0 * refract(vec3(0.0, 0.0, -1.0), sphereNormal, refractiveIndex); refractedVector.xy = -refractedVector.xy; highp vec3 finalSphereColor = texture2D(inputImageTexture, (refractedVector.xy + 1.0) * 0.5).rgb; // Grazing angle lighting highp float lightingIntensity = 2.5 * (1.0 - pow(clamp(dot(ambientLightPosition, sphereNormal), 0.0, 1.0), 0.25)); finalSphereColor += lightingIntensity; // Specular lighting lightingIntensity = clamp(dot(normalize(lightPosition), sphereNormal), 0.0, 1.0); lightingIntensity = pow(lightingIntensity, 15.0); finalSphereColor += vec3(0.8, 0.8, 0.8) * lightingIntensity; gl_FragColor = vec4(finalSphereColor, 1.0) * checkForPresenceWithinSphere; }

So what this calculation does is it adjusts for a non-square aspect ratio of the image. The image aspect ratio is passed in as a uniform, and this adjusts the normally 0.0-1.0 texture coordinate for the Y axis to instead be from 0.0-1.0*(imageHeight/imageWidth). It has to expand the Y axis coordinate about its center point (0.5), thus the weird addition and subtraction in there. If you don’t do this, your sphere turns into an egg in non-square images.

highp vec2 textureCoordinateToUse = vec2(textureCoordinate.x, (textureCoordinate.y * aspectRatio + 0.5 - 0.5 * aspectRatio));

So what this calculation does is it adjusts for a non-square aspect ratio of the image. The image aspect ratio is passed in as a uniform, and this adjusts the normally 0.0-1.0 texture coordinate for the Y axis to instead be from 0.0-1.0*(imageHeight/imageWidth). It has to expand the Y axis coordinate about its center point (0.5), thus the weird addition and subtraction in there. If you don’t do this, your sphere turns into an egg in non-square images.

highp float distanceFromCenter = distance(center, textureCoordinateToUse);

This is a Pythagorean distance calculation to determine how far the current pixel (texture coordinate) is from the center that we’ve provided as a uniform. It’s a sqrt(xdiff^2 + ydiff^2) calculation.

lowp float checkForPresenceWithinSphere = step(distanceFromCenter, radius);

The step() function returns 1 if the second value is greater than the first, 0 if not. I use these to avoid if() statements, which are expensive in fragment shaders (branching does not work well in massively parallel operations).

distanceFromCenter = distanceFromCenter / radius;

This normalizes the distance from the center to be 0.0 for a value at the center of the sphere, and 1.0 for a value at the edge of the sphere. Values outside that range will get filtered out by the product of the above step() calculation later on.

highp float normalizedDepth = radius * sqrt(1.0 - distanceFromCenter * distanceFromCenter);

This is where we calculate the Z height that a sphere, cut in half, would extend above the plane of the image. This is another geometrical calculation based on knowing the distance of our point from the center of the sphere.

highp vec3 sphereNormal = normalize(vec3!(textureCoordinateToUse - center, normalizedDepth));

Once we know the height of the spherical cap at that point, we can calculate the normal for that point on the sphere’s surface. Think of it as a ray that extends from the center of the sphere to the surface at this X, Y coordinate. The normal is based on the center of the sphere, so we subtract our aspect-ratio-adjusted-coordinate from the center to get the relative X, Y coordinate from the center of the sphere. The Z component is the height of the sphere we just calculated.

highp vec3 refractedVector = refract(vec3(0.0, 0.0, -1.0), !sphereNormal, refractiveIndex);

With the normal, we can calculate the refraction of light from that point on the sphere’s surface. The refraction calculation uses the surface normal, the refractiveIndex (a material property that you pass in, I think I use glass’s here), and a ray direction. I believe the ray direction here is from the eye going into the screen, although I can never remember positive/negative Z directions in OpenGL. This then generates a refracted vector, pointing in the direction light would as it refracts through a sphere.

gl_FragColor = texture2D(inputImageTexture, (refractedVector.xy + 1.0) * 0.5) * checkForPresenceWithinSphere;

We then take this refracted vector, which is in the -1.0-1.0 coordinate space, and adjust it to the 0.0-1.0 coordinate space for texture sampling. We read the texture color at the location pointed to by that vector. The earlier step() function to determine if a point was within the sphere comes into play here, where we only display a color if the point was within the sphere. If it was not, we output 0.0, 0.0, 0.0, 0.0 as an RGBA color because that’s the result of multiplying with 0.

The End

the day you finally use algebra: a 3d math primer

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