Computer GraphicsCamera Projection / Picking

CO2409

Week 8 - Optional Advanced Material

Not on Exam

ContentsContents

1. World / View Matrices Recap2. Projection Maths3. Pixel from World-Space Vertex4. World Space Ray from Pixel

• These notes are presented as additional material and are not examinable

Model SpaceModel Space

• An model’s mesh is defined in its own local coordinate system - model space

• Each model is positioned with a matrix– Transforming it from model

space into world space

• This matrix is called the World Matrix

World to Camera SpaceWorld to Camera Space

• Next consider how the models are positioned and oriented relative to the camera

• Convert the models from world space into camera space– The scene as viewed from

camera’s position

• This transformation is done with the view matrix

Camera to Viewport SpaceCamera to Viewport Space

• Finally project the camera space models into 2D

• The 3D vertices are projected to camera position

• Assume the viewport is an actual rectangle in the scene

• Calculate where the rays hit the viewport = 2D geometry

• This is done with the projection matrix

Projection DetailsProjection Details

• Cameras have two settings:– Field of View (FOV)– Viewport distance (D)

• Viewport distance is same as the near clipping plane– Where geometry “slices”

through the viewport

• FOV works as a wide angle or zoom lens– FOV can be different for width

and height – FOVX & FOVY

Projecting a VertexProjecting a Vertex

• Consider the projection of a single 3D vertex to 2D

• Want 2D coordinates (XV, YV)– YV not shown in diagram

• Calculate using similar triangles:X / Z = XV / D, so XV = D * X / Z

In a similar way, YV = D * Y / Z This is the perspective divide

• Now have 2D coords, but still in camera space units– Need to convert to pixels

Converting to PixelsConverting to Pixels

• Calculate the actual viewport dimensions (camera space): tan(FOVX / 2) = (WV / 2) / D

so WV = 2 * D * tan(FOVX / 2)

similarly, HV = 2 * D * tan(FOVY / 2)

• Then calculate:XN = 2 * XV / WV

YN = 2 * YV / HV

• This 2D coordinate(XN, YN) is in the range –1 to 1– Ready to convert to pixel position

Converting to PixelsConverting to Pixels

• If the viewport width & height (in pixels) are WP and HP:then XP= (XN + 1) WP / 2

and YP= (1 - YN) HP / 2– The second formula flips the Y

axis (viewport Y is down)

• (XP, YP) are the coordinates of the final pixel we want

PickingPicking

• Sometimes we need to manually perform the projection process:– To find the pixel for a particular 3D point– E.g. To draw text/sprites in same place as a 3D model

• Or perform the process in reverse:– Each 2D pixel corresponds to a ray in 3D space (refer

to the projection diagram)– Finding and working with this ray is called picking– E.g. to find the 3D object under the mouse

• The algorithms for both follow – they are derived from the previous slides

Pixel from World-Space VertexPixel from World-Space Vertex

• Start with world space vertex P• Transform this vertex by combined view / projection

matrix to give Q• If Q.z < 0 then the vertex is behind us, discard• Otherwise do perspective divide:

Q.x /= Q.z and Q.y /= Q.z

• Finally, scale to pixel coordinates X,Y:X = (Q.x + 1) * (ViewportWidth / 2)Y = (1 - Q.y) * (ViewportHeight / 2)

• Use to draw text/sprites in same place as 3D entity

World-Space Ray From Pixel 1World-Space Ray From Pixel 1

• Initial pixel (X,Y), first convert to point Q in the range -1 -> 1:Q.x = (2 * X / ViewportWidth) - 1

Q.y = 1 – (2 * Y / ViewportHeight)

• Set Q.z = D (viewport / near clip distance)– The result vertex will be exactly on the clip plane

• Calculate viewport size in camera space:WV = 2 * D * tan(FOVX / 2)

HV = 2 * D * tan(FOVY / 2)

• If FOVY not available:

HV = WV * ViewportHeight / ViewportWidth

World-Space Ray From Pixel 2World-Space Ray From Pixel 2

• Convert Q into camera space:Q.x *= WV / 2

Q.y *= HV / 2

• Finally transform by the inverse of the view matrix to get a point in world space

• Then cast a ray from camera to this point– Use this 3D ray to detect the entity at the pixel