Basic 3D Concepts. Overview 1.Coordinate systems 2.Transformations 3.Projection 4.Rasterization

Basic 3D Concepts

Basic 3D Concepts

Overview1. Coordinate systems

2. Transformations

3. Projection

4. Rasterization

ESTABLISHING A COORDINATE SYSTEM

Representing the 3D world Typically, objects in our world consist

of groups of triangles.

face = set of one or more contiguous coplanar adjacent triangles. adjacency?

How do we represent triangles?

Representing the 3D world

How do we represent triangles? By 3 points - the 3 vertices of the

triangle.

How are the points represented?

Representing the 3D world

How are the points represented? Since we are in 3D space, each point is a vector

consisting of 3 values, <x,y,z>, in a cartesian coordinate system. (scalar, vector, matrix)

2D 3D

Representing the 3D world In TorqueScript, a vector is

represented by a list/string of (typically) 3 numbers separated by a space. Example

$fred = “12.0 13 19”;$c0 = getword( $fred, 0 );echo( $c0 ); //what does this print?

Representing the 3D world Unity, uses the Vector3 class.

example (JavaScript)var aPosition = Vector3(1, 1, 1);

example (C#)using UnityEngine;using System.Collections;

public class example : MonoBehaviour {public Vector3 aPosition = new Vector3( 1, 1, 1 );

}

Unity’s Vector3 class

This structure is used throughout Unity to represent D positions and directions.

It also contains functions for doing common vector operations.

See http://unity3d.com/support/documentation/ScriptReference/Vector3.html for more information.

Unity’s Vector3 class

class variables: one, zero, forward, up, right

instance variables: x, y, z

methods: scale, normalize, cross, dot, reflect,

distance, etc. operators:

+, -, *, /, ==, !=

Representing the 3D world How does a vector such as “12.0 1 -5” map

into the “real” world?

We don’t (yet) know if 1 above specifies the H, W, or D!

Furthermore, we don’t know the relationship (l or r, u or d, f or b) between

“12.0 1 -5” and “12.0 2 -5”

Representing the 3D world How does a vector map into the “real”

world? Let’s establish a coordinate system Left is left-handed; right is right-handed. Index is +z, thumb is +y, middle is +x.

Representing objects Objects (models) are composed of polygons

which are composed of triangles.

But these triangles aren’t arbitrary!

Representing objects

Objects (models) are composed of polygons which are composed of triangles.

But these triangles aren’t arbitrary!

Representing the 3D world

We don’t know where our object will be placed in the world. It may even move in the world! We may have more than one in the world

too! So we don’t/can’t fix the object

coordinates in terms of world coordinates!

But we need to specify each of the triangle vertices as numbers.

So what can we do?

Representing the 3D world

So we don’t/can’t fix the object coordinates in terms of world coordinates!

But we need to specify each of the triangle vertices as numbers.

So what can we do? Each object has it’s own object

coordinate system with it’s own origin (0,0,0).

So where is the model origin?

Representing the 3D world

Each object has it’s own object coordinate system with it’s own origin (0,0,0).

So where is the model origin? Anywhere! It can be any point on (or

not on) the object.1. It can be the left-most (or right-most or

top-most or …) point on the object.2. It can be the geometric center

(centroid/center of mass) of the object.

Representing the 3D world

Typically, it is the geometric center (centroid/center of mass) of the object.

Let P be the set of points in the object.

Let Pi=<xi,yi,zi> be a particular point.

How can we calculate the centroid of an object?

Representing the 3D world Typically, it is the

geometric center (centroid/center of mass) of the object. Let P be the set of

points in the object.

Let Pi=<xi,yi,zi> be a particular point.

How can we calculate the centroid of an object?

P

z

P

y

P

xzyx

P

ii

P

ii

P

ii

111 ,,,,

Representing the 3D world

Interesting property of the centroid:

This may not even be a point on the object!

Can you think of a real world object with a center that isn’t on the object?

TRANSFORMATIONS

Transformation

Conversion from object coordinates to world coordinates. Consists of:

Rotation Translation Scale

Transformation

Translation

Transformation

Scale

Transformation

Rotation

Representing the 3D world

In Unity, the Transform Component determines the actual Position, Rotation, and Scale of all objects in the scene.

Every object has a Transform.

Representing the 3D world

In Unity, … Position

X, Y, and Z coordinates Rotation

around the X, Y, and Z axes, measured in degrees

Scale along X, Y, and Z axes A value "1" is the original size (size at which

the object was imported).

Representing the 3D world

In Unity, … All properties of a Transform are

measured relative to the Transform's parent.

If the Transform has no parent, the properties are measured relative to World Space.

PROJECTION

Problem

We have a 3D world consisting of height, width, and depth which is displayed on a 2D computer screen consisting only of height and width!

Projection

Remember that our world is 3D and the computer monitor is 2D.

Projection is the conversion from world coordinates to screen coordinates. Types:

1. parallel (orthographic)2. perspective

Parallel (orthographic) projection

The distance from the camera doesn't affect how large an object appears.

Perspective projection

The further an object is from the camera (viewpoint), the smaller it appears.

Similar to how our eye and how a camera works.

Controlling perspective projection

In both scenes, the fence appears to be relatively close, but the mountains vary greatly.

Controlling perspective projection

Controlling perspective projection

Entire transformation process

RASTERIZATION

Rendering (rasterization)

But that’s not all there is to do! All we’ve done so far is to project the

vertices of triangles onto the screen. What about the points in between (the

vertices)? What about color? What about light sources?

Rendering (rasterization)

The process of converting the 3D model of an object into an on-screen 2D image.

Note: This is most often done by the video card hardware for speed.

Rendering examples

Rendering examples

Note uniformity across face of triangle.

Rendering

Steps:1. transformation

2. projection

3. scan conversion (filling in the triangles) involves shading the surface (considers the

orientation of the surface w.r.t. the location of the light(s))

Rendering

Typically involves a z-buffer (because many points may be projected to the same point on the screen).

PHEW!