3D Projections: 3D Projections Display device (a screen) is 2D… 3D to 2D is more complicated… Solution : Transform 3D objects on to a 2D plane using projections PowerPoint Presentation: Normalization transformation Modeling transformation Modeling coordinates Viewing transformation Viewing coordinates Projection transformation Projection coordinates Normalized projection coordinates Device coordinates Workstation transformation World coordinates Types of Projections: Types of Projections 2 types of projections Perspective Parallel In parallel projection , coordinate positions are transformed to the view plane along parallel lines. In perspective projection , object position are transformed to the view plane along lines that converge to a point called projection reference point (center of projection) PowerPoint Presentation: Parallel Projection Perspective Projection PowerPoint Presentation: PROJECTIONS PARALLEL (parallel projectors) PERSPECTIVE (converging projectors) One point (one principal vanishing point) Two point (Two principal vanishing point) Three point (Three principal vanishing point) Orthographic (projectors perpendicular to view plane) Oblique (projectors not perpendicular to view plane) General Cavalier Cabinet Multiview (view plane parallel to principal planes) Axonometric (view plane not parallel to principal planes) Isometric Dimetric Trimetric Perspective v Parallel: Perspective v Parallel Perspective: visual effect is similar to human visual system... has 'perspective foreshortening' size of object varies inversely with distance from the center of projection. Projection of a distant object are smaller than the projection of objects of the same size that are closer to the projection plane. Parallel: It preserves relative proportion of object. less realistic view because of no foreshortening however, parallel lines remain parallel. Perspective Projections: Perspective Projections When a 3D object is projected onto view plane using perspective transformation equations, any set of parallel lines in the object that are not parallel to the projection plane, converge at a vanishing point. There are an infinite number of vanishing points, depending on how many set of parallel lines there are in the scene. If a set of lines are parallel to one of the three principle axes, the vanishing point is called an principal vanishing point . There are at most 3 such points, corresponding to the number of axes cut by the projection plane. PowerPoint Presentation: Vanishing points Certain set of parallel lines appear to meet at a different point The V anishing point for this direction Principal vanishing points are formed by the apparent intersection of lines parallel to one of the three principal x, y, z axes. The number of principal vanishing points is determined by the number of principal axes intersected by the view plane. Sets of parallel lines on the same plane lead to collinear vanishing points. The line is called the horizon for that plane Perspective Projections: Perspective Projections 2 different examples of a one-point perspective projection of a cube. (note: x and y parallel lines do not converge) Perspective Projections: Perspective Projections Two-point perspective projection: This is often used in architectural, engineering and industrial design drawings. Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection. Parallel Projections: Parallel Projections We can define a parallel projection with a projection vector that defines the direction for the projection lines. 2 types: Orthographic : when the projection is perpendicular to the view plane. In short, direction of projection = normal to the projection plane. the projection is perpendicular to the view plane. Oblique : when the projection is not perpendicular to the view plane. In short, direction of projection normal to the projection plane. Not perpendicular. PowerPoint Presentation: Orthographic projection Oblique projection when the projection is perpendicular to the view plane when the projection is not perpendicular to the view plane Orthographic (or orthogonal) projections:: Orthographic (or orthogonal) projections: Front, side and rear orthographic projection of an object are called elevations and the top orthographic projection is called plan view . all have projection plane perpendicular to a principle axes. Here length and angles are accurately depicted and measured from the drawing, so engineering and architectural drawings commonly employee this. However, As only one face of an object is shown, it can be hard to create a mental image of the object, even when several views are available. PowerPoint Presentation: Orthogonal projections: Axonometric orthographic projections: Axonometric orthographic projections Orthographic projections that show more than one face of an object are called axonometric orthographic projections. The most common axonometric projection is an isometric projection where the projection plane intersects each coordinate axis in the model coordinate system at an equal distance. PowerPoint Presentation: Cavalier projection : All lines perpendicular to the projection plane are projected with no change in length. 2 common oblique parallel projections: Cavalier and Cabinet PowerPoint Presentation: Cabinet projection : Lines which are perpendicular to the projection plane (viewing surface) are projected at 1 / 2 the length . This results in foreshortening of the z axis, and provides a more “realistic” view.