아이폰 3d 프로그래밍 챕터2

iPhone 3DProgramming

Chapter 02. Math and Metaphors

2012. 01. 26.Sanghoon Lee

Sunday, December 9, 12

수학과 비유

Graphics Rendering Pipeline & Metaphor

Transforms

Matrix stack

Vector Beautification with C++

HelloCone


What we’re going to study!


Different kind of topology


Different kind of topology

The first argument specifies the topology


Drawing rectangle


Properties with Vertices


Setting vertex properties examples


Homogeneous coordinates

Internally, the OpenGL implementation always converts it into a 4D floating-point number.

It’s an artificial construction that allows all transformations to be represented with matrix multiplication.

These 4D coordinates are known as homogeneous coordinates.

So, shortly after entering the assembly line, all vertex positions become 4D; don’t they need to become 2D at some point? The answer is yes.


The Life of a Vertex

Perspective transform ( removal of w )


Vertex goes from 4D to 2D

Early life of a vertex. Top row is conceptual; bottom row is OpenGL’s view


Vertex goes from 4D to 2D


The photography metaphor

1. Arrange the various dishes on the table.2. Arrange one or more light sources.3. Position the camera.4. Aim the camera toward the food.5. Adjust the zoom lens.6. Snap the picture.

1. Adjust the camera’s field-of-view angle. (projection matrix)2. Position the camera and aim it in the appropriate direction. (view matrix)3. For each object:

a. Scale, rotate, and translate. (model matrix)b. Render the object.


ES 1.1

ES 2.0


Matrix Multiplication


Model Matrix

Scale

Translation

Rotation

The three most common operations when positioning an object in a scene


Setting the Model Matrix (Scale)


Setting the Model Matrix (Translation)


Setting the Model Matrix (Rotation ES 1.1)

glRotatef(m_currentAngle, 0, 0, 1);

counter-clockwise rotation about the Z-axis

angle in degrees

the latter three arguments define the axis of rotation


Setting the Model Matrix (Rotation ES 2.0)


Setting the Model Matrix (Rotation)

arbitrary axis rotation* ES 1.1

glRotatef generates the matrix for youglRotatef only rotates about the origin


Setting the Model Matrix (Rotation)

arbitrary axis rotation* ES 2.0


What we have been through


Setting the View Transform


Setting the View Transform

The simplest way to create a view matrix


Setting the Projection Transform

think of the projection as being the camera's "zoom lens"



ES 1.1



ES 2.0

OrthographicPerspective


Matrix stacks


Matrix stacks

Only ES 1.1 supportsYou need to create by yourself in ES 2.0


Interpolations


Example of easing equations


Quaternions

For position keyframes and color keyframes

What if interpolating between two orientations?

It's easy


Quaternions

Storing an angle for each joint would be insufficient

you'd also need the axis of rotation

This is known as Axis-Angle notation

requires a total of four floating-point values for each joint


Quaternions





Quaternions


Quaternions





Study and understand it!!


Vector Beautification template <typename T>struct Vector3 {

Vector3() {}Vector3(T x, T y, T z) : x(x), y(y), z(z) {}void Normalize(){

float length = std::sqrt(x * x + y * y + z * z);x /= length;y /= length;z /= length;

}

Vector3 Normalized() const{

Vector3 v = *this;v.Normalize();return v;

}

Vector3 Cross(const Vector3& v) const{

return Vector3(y * v.z - z * v.y,z * v.x - x * v.z,x * v.y - y * v.x);

}T Dot(const Vector3& v) const{

return x * v.x + y * v.y + z * v.z;}Vector3 operator-() const{

return Vector3(-x, -y, -z);}

bool operator==(const Vector3& v) const{

return x == v.x && y == v.y && z == v.z;}

T x;T y;T z;

};


Vector Beautification

vec3 z;normalize(&z, &(eye-target));

vec3 cross;cross(&cross, &up, &z);vec3 x;normalize(&x, &cross);

cross(&cross, &up, &z);vec3 y;normalize(&y, &cross);

vec3 z = (eye - target).Normalized();

vec3 x = up.Cross(z).Normalized();

vec3 y = z.Cross(x).Normalized();


Hello ConeFixed Function


Hello ConeShaders


Further study

Graphics Rendering pipeline

Quaternion


Q/A


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