Viewpoints and Transformations CSIS 5838: Graphics and Animation for Gaming

Coordinate Systems Object Space: Coordinate space of an individual mesh – Edit mode in Blender Translating/rotating/scaling in this space changes location of individual vertices in this coordinate system – Usually with respect to some “object center” Howard Hamilton, University of Regina

Coordinate Systems World Space: Overall coordinate space of all elements – Object mode in Blender Translating/rotating/scaling object in this space changes object space relative to world space – “Object center” manipulated – All vertices move with center Howard Hamilton, University of Regina

Coordinate Systems Image Space: Coordinate system from POV of viewer/camera – Camera view in Blender Convention: Center of Interest (COI) along z-axis in this space Moving camera/viewpoint changes this space relative to world space Howard Hamilton, University of Regina

Affine Transformations Transformation between coordinate systems using matrix multiplication General affine transformation of point (x, y, z) in one coordinate system to (x’, y’, z’) in another coordinate system:

Translation and Scaling Translation by [p, q, r]: Scaling by [p, q, r]:

Rotation About Different Axes By α around x-axis: By α around z-axis:

Combined Transformations Can combine all transformation into single matrix with multiplication M obj2world = M objtrans x M objscale x M objrotX x M objrotY x M objrotZ M world2eye = M worldtrans x M worldscale x M worldrotX x M worldrotY x M worldrotZ M obj2eye = M obj2world x M world2eye Efficiency: – Compute M obj2world for each object – Compute M world2eye once for entire world – Compute M obj2eye for each object – Apply M obj2eye to each vertex in each object

The Rendering Pipeline Conversion of vertices, etc. on mesh in object space to world space Projection of points in 3D world space into 2D image space Other modifiers – Surface features (colors, UV images etc.) – Transformation of apparent surfaces (smoothing, etc.) –…–…

Visual Frustum Area of world rendered to screen (“field of view”) Near clip plane = image plane (where image “projected”) Far clip plane = limit of view – Nothing further from camera rendered – Also nothing outside of “cone” rendered

Orthographic vs. Perspective View Projecting vertices in 3D image space to 2D image plane Orthographic view: (x, y, z) in image space  (x, y) on image plane – No foreshortening – Makes editing objects simpler blender.org

Perspective View “Normal” vision with foreshortening Image projected to point representing location of viewer Focal length f = distance between viewer and image plane – Smaller f = more foreshortening blender.org

Perspective Transformation Foreshortening: (x, y, z) in image space  (xf/z, yf/z) on image plane gamedev.com