Projections PDF

Projections PROJECTION Projection can be defined as a mapping of point P(x,y,z) into its image P’(x’,y’,z’) in the projection plane which constitute the display surface. In short it is the transformation of a three- dimensional (3D) object into a two- dimensional (2D) plane.  Classificatio...

Projections PROJECTION Projection can be defined as a mapping of point P(x,y,z) into its image P’(x’,y’,z’) in the projection plane which constitute the display surface. In short it is the transformation of a three- dimensional (3D) object into a two- dimensional (2D) plane.  Classification Projection Perspective Parallel Projection Projection Orthograph Oblique One ic Point Axonometri General Two c Multi-view Points Isometri c Three Di- Cavalie Points metric Cabinet r Trimetri c PARALLEL  PROJECTION In this, coordinate positions are transformed to the view plane along parallel lines. Projection lines are parallel to each other. Projection lines are extended from the object and intersect the view plane. 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. 6 Orthographic Oblique projection projection when the projection when the projection is is perpendicular to not perpendicular to the the view plane view plane 7 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. 8 Parallel Projections Projectors parallel. COP at infinity. 9 Standard Orthographic Projection  xp  1 0 0 0  x        yp    0 1 0 0  y  zp  0 0 0 0  z        1   0 0 0 1  1  Parallel projections: summary Center of projection is at infinity. Projectors are parallel. Parallel lines stay parallel There is no forshorthening Distances and angles are transformed consistently Used most often in engineering design, CAD systems. Used for top and side drawings from which measurements could be made. Multi-View Orthographic Projections Orthographic projections that show only one side of an object are called multi- view orthographic projections When direction of projection is parallel to any principal axis, this produces front, top and side view of an object Orthographic Projection: projectors orthogonal to projection plane same for all points DOP (direction of projectors) 13 Orthographic Projections DOP is perpendicular to the view plane Orthogonal projections: 15 Orthographic Projections  Orthographic Projections are a collection of 2- D drawings that work together to give an accurate overall representation of an object. Defining the Six Principal Views or Orthographic Views Which Views to Present? General Guidelines Pick a Front View that is most descriptive of object Normally the longest dimension is chosen as the width (or depth) Most common combination of views is to use: Front, Top, and Side View 2.OBLIQUE:  Projection lines are parallel to each other but not perpendicular to view plane.  In this there are 2 types: i. CAVALIER:-In this,projection lines makes angle of 30degree with the view plane & there is no change in the length of projection. ii. CABINET:-In this,projection lines makes angle of 45degree with the view plane & length of projected line vil reduce. Oblique Parallel Projections Most general parallel views Projectors make an arbitrary angle with the projection plane Angles in planes parallel to the projection plane are preserved Oblique Projection degree of obliqueness  projectors are not orthogonal to image plane avalier ngle between projectors and projection plane is 45°. Lines orthogonal to the projection p etain their exact length. Perpendicular faces are projected at full scale Cabinet Angle between projectors and projection plane is arctan(2)=63.4°. Lines orthogonal to th projection plane are projected at half length. Perpendicular faces are projected at 50% sc Looks like forshorthening.  2 common oblique parallel projections: Cavalier and Cabinet Cavalier projection: All lines perpendicular to the projection plane are projected with no change in length. 28 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. 29 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. 30 Axonometric Projection Orthographic projections that show more than 1 side of an object are called axonometric orthographic projections In axonometric orthographic projections, the direction of projection is not parallel to any of the principal axis. ….. Sub categories of Axonometric orthographic Projections are; 1. Isometric 2. Di-metric 3. Trimetric Continued….. Isometric Projection: The direction of projection makes equal angles with all of the principal axis. Isometric Projection Projector makes equal angles with all three principal axes All three axes are equally foreshortened Mechanical Drawing isometric Continued….. Di-metric Projection: The direction of projection makes equal angles with exactly two of the principal axis. Continued….. Trimetric Projection: The direction of Projection makes unequal angles with the principal axis. 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. 38 Perspective Projections Characteristics: Center of Projection (CP) is a finite distance from object Projectors are rays (i.e., non-parallel) Vanishing points Objects appear smaller as distance from CP (eye of observer) increases Difficult to determine exact size and shape of object Most realistic, difficult to execute 39  Perspective Projection Most natural for people In human vision, perspective projection of the world is created on the retina (back of the eye) Used in CG for creating realistic images Perspective projection images carry depth cues Foreshorthening causes distant objects to appear smaller Relative lengths and angles are not preserved A perspective image cannot be used for metric measurements of the 3D world Parallel lines not parallel to the image plane converge at a vanishing point An axis (principal) vanishing point is a point of convergence for lines parallel to a principal axis of the object. We distinguish one-, two-, three-point projections.) Viewing, projections Hofstra University 40 By Altaf Ahmed, email: the.nothingn Vanishing Points 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. 43  Vanishing points Certain set of parallel lines appear to meet at a different point – The Vanishing 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 44 Classes of Perspective Projection  One-Point Perspective  Two-Point Perspective  Three-Point Perspective 45 One point : This perspective projection occurs when the projection plane is perpendicular to one of the principle axis Two points: this perspective projection occurs when the projection plane intersects exactly two of the principal axis. Three points: this perspective projection occurs when the projection plane intersect all the three principal axes, x,y and z axes. By Altaf Ahmed, email: the.nothingn Types of Perspective Projection THERE ARE 3 TYPES OF PERSPECTIVE PROJECTION: 1.ONE-POINT PERSPECTIVE PROJECTION Two-point perspective projection: This is often used in architectural, engineering and industrial design drawings.  49 Vanishing Points Viewing, projections Hofstra University 50 Three-point perspective projection Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection 51 2.TWO-POINT PERSPECTIVE PROJECTION 3.THREE-POINT PERSPECTIVE PROJECTION