3D Concepts UNIT 3

3-D Coordinate Spaces Remember what we mean by a 3-D coordinate space x axis y axis z axis P y z x Right-Hand Reference System

Translations In 3-D To translate a point in three dimensions by tx, ty and tz simply calculate the new points as follows: x’ = x + tx y’ = y + tyz’ = z + tz (x’, y’, z’) (x, y, z) Translated Position

Scaling In 3-D To scale a point in three dimensions by sx, sy and sz simply calculate the new points as follows: x’ = sx*xy’ = sy*yz’ = sz*z (x, y, z) Scaled Position (x’, y’, z’)

Rotations In 3-D x’ = x·cosθ - y·sinθ y’ = x·sinθ + y·cosθ z’ = z x’ = x y’ = y·cosθ - z·sinθ z’ = y·sinθ + z·cosθ x’ = z·sinθ + x·cosθ y’ = y z’ = z·cosθ - x·sinθ The equations for the three kinds of rotations in 3-D are as follows:

Homogeneous Coordinates In 3-D Similar to the 2-D situation we can use homogeneous coordinates for 3-D transformations - 4 coordinate column vector All transformations can then be represented as matrices x axis y axis z axis P y z x P (x, y, z) =

3D Transformation Matrices Translation by tx, ty, tz Scaling by sx, sy, sz Rotate About X-AxisRotate About Y-AxisRotate About Z-Axis

Projections Our 3-D scenes are all specified in 3-D world coordinates To display these we need to generate a 2-D image - project objects onto a picture plane So how do we figure out these projections? Picture Plane Objects in World Space

Converting From 3D To 2D Projection is just one part of the process of converting from 3D world coordinates to a 2D image Clip against view volume Project onto projection plane Transform to 2-D device coordinates 3-D world coordinate output primitives 2-D device coordinates

3D Viewing In 2D viewing we have 2D window & 2D viewport & objects in the world coordinates. The 3D viewing has an added dimension which makes it complex as even though objects are 3D the display devices are only 2D. The mismatch between 3D objects & 2D displays is compensated by introducing projections. The projection transforms 3D objects into a 2D projection plane. View plane: It is nothing but the film plane in a camera which is positioned & oriented for a particular shot of the scene. World coordinates positions in the scene are transformed to viewing coordinates, then viewing coordinates are projected onto the view plane.

View reference point: This point is the center of our viewing coordinate system. The production of a 2D image of higher dimensional object refers to graphical projection. A projection can be defined as a mapping of any point P[x,y,z] to its image P`[x`,y`,z`] onto the view plane, called as projection plane. Parallel & perspective projections are the two broad categories of projections.

Types of Projections There are two broad classes of projection: – Parallel: Typically used for architectural and engineering drawings. – Perspective: Realistic looking and used in computer graphics. Perspective Projection Parallel Projection

Parallel Projections Some examples of parallel projections Orthographic Projection Isometric Projection

Perspective Projections Perspective projections are much more realistic than parallel projections

Geometric projection Perspective Parallel Orthogr aphic Three- point Two- point Oblique To p One- point Other Cavalier Isometric Cabinet Fro nt Ax on om etr ic Side elev atio n

Parallel projection: – If the direction of projection is perpendicular to the projection plane, it is an orthographic projection. – If the direction of projection is not perpendicular to the projection plane is called as oblique projection. – A multi-view projection displays a single face of a 3D object. – Axonometric projections allow the user to place the view-plane normal in any direction such that 3 adjacent faces of a cube like object are visible. – Dimetric projections differ from isometric projections in the direction of the view-plane normal.

– Trimetric projections allow the viewer the most freedom in selecting the components of n. Perspective projection: – It is a type of projection where 3D objects are not projected along parallel lines, but along lines emerging from a single point. – A vanishing point is a point in a perspective drawing to which parallel lines appear to converge. – One-point perspective exists when a painting plate is parallel to two axes of a rectilinear scene. – Two point perspective

Assignment Orthographic Wireframe Elevation Orthographic Wireframe Plan Orthographic Wireframe End-Elevation Perspective View