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Parallel Projections

Parallel Projections. Parallel projections are defined by a Direction Of Projection (DOP) and a projection plane DOP also called projection vector Coordinate positions are transformed to the view plane along parallel lines Important property: they preserve relative proportions of objects

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Parallel Projections

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  1. Parallel Projections • Parallel projections are defined by a Direction Of Projection (DOP) and a projection plane • DOP also called projection vector • Coordinate positions are transformed to the view plane along parallel lines • Important property: they preserve relative proportions of objects • Less realistic effect

  2. Parallel projections (projection plane) DOP

  3. Parallel Projections • Classified as orthographic or oblique • The DOP makes 2 angles with the projection plane • Orthographic means DOP is perpendicular to the projection plane, i.e. both angles are 90 degrees • Oblique means DOP not perpendicular • i.e. one or both angles are not equal to 90 degrees DOP DOP orthographic oblique

  4. Parallel projections • 3 parallel projections of an object showing relative proportions from different viewing positions. • Used in engineering and architectural drawings: object represented through a set of views => its appearance can be reconstructed from these views

  5. Plan Elevation Model Side Elevation Front Elevation Orthographic projections

  6. Orthographic projections Orthographic projections of an object (plan + elevation views)

  7. Oblique • Oblique projections are used togive a “flat” view of the most important side and view of two other sides at a given angle f (e.g., 45 degrees) • This view is most useful if we need to show a small detail on one side but don't need allother complete views (e.g., plan and elevation). f

  8. View Plane COP Perspective Projections - 1 • The larger-sized object will be smaller on the view-plane becauseit is further away from the Centre of Projection (COP)

  9. Perspective Projections - 2 • To obtain a perspective projection of a three-dimensional object, we transform points along projection lines that meet at the COP • Example: z=d projection plane; (0,0,0) COP y-axis P(x,y,z) x-axis PP(xP,yP,d) z-axis d

  10. Projection Plane x-axis xP P(x,y,z) z-axis d d z-axis P(x,y,z) y-axis Other Views • View along y axis • View along x axis Projection Plane

  11. Projection Plane x-axis xP x xP = d z P(x,y,z) yP y = d z z-axis d d.x x d.y y yP = = z z/d z z/d Perspective projections: equations • To calculate PP = (xP,yP,zP), the perspective projection of (x,y,z) onto the projection plane at z = d we use the shared properties of similar triangles • Mutiplying each side by d yields xP = = • The distance d is just a scale factor applied to xP and yP • All values of z are allowable except z = 0

  12. x , y , z/d z/d Homogeneous Matrix Form • Using a 3D homogeneous coordinate representation, we can write the perspective projection transformation matrix form x y z 1 Xp Yp Zp 1 d/z 0 0 0 0 d/z 0 0 0 0 d/z 0 0 0 1/z 0 = ( ) = (xP, yP, zP, 1) d, 1

  13. Projection Equations Cont. • This makes sense intuitively: the further away from the origin (our COP) a point P is, the larger its z value • By dividing the x and y co-ordinates of every point of an object by the z coordinate means that objects further away will have each x and y divided by a larger number, and therefore the projection onto the projection plane will be much smaller than objects that are closer to the COP (in this case the origin)

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