Basic 3D Concepts

Overview 1.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,, in a cartesian coordinate system. (scalar, vector, matrix) 2D3D

Representing the 3D world  In TorqueScript, a vector is represented by a list/string of (typically) 3 numbers separated by a space. Example $fred = “ ”; $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 tation/ScriptReference/Vector3.html for more information. tation/ScriptReference/Vector3.html

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 “ ” 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 “ ” and “ ”

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 P i = 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 P i = be a particular point. How can we calculate the centroid of an object?

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

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

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!