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III. Projections
Imagephysics
Old definition
Definitions3D objects to 2D imageSolution Rn to Rm, m<n through a projectionThe projection lines (rays, projectors) cross a given point of the space, named projection center (viewing point) and each point of the object are intersecting the projection plane to form the projection.
Classification
Display = plate surface => plane gemetricalsurfaces? D = distance (viewing point, projection plane)
(a) D = infinity: parallel projections(b) D< infinity: perspective projections
Classification
(a) Perspective projection: plan & center of projection(b) Parallel projection: plan & projection direction
Other than planar projections
Why to use each?
Parallel projections used for engineering and architecture because they can be used for measurementsPerspective imitates eyes or camera and looks more natural
|| lines can convergeObjects farther away are more foreshortened (i.e., smaller) than closer ones
Further classification
Examples
History
Pespective projection
For right-angled forms whose face normalsare perpendicular to the x, y, z coordinate axes, number of vanishing points = number of principal coordinate axes intersected by projection planeOne Point Perspective (z-axis vanishing point)
y
x
y
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z
z
Two and three point perspectives
Two Point Perspective(z, and x-axis vanishing points)Three Point Perspective(z, x, and y-axis vanishing points)
y
xz
y
x
z
y
z
x
y
xz
Perspective Projection
Used for:advertisingpresentation drawings for architecture, industrial design, engineeringfine art
Pros:gives a realistic view and feeling for 3D form of object
Cons:does not preserve shape of object or scale (except where object intersects projection plane)
Different from a parallel projection becauseparallel lines not parallel to the projection plane convergesize of object is diminished with distanceforeshortening is not uniform
Camera
Camera
Perspective projection
Perspective projection
One point perspective
One point perspective
Two point perspective
Three point Perspective
Perspective projection
Assume perspective proj. on plane z=0, with 1 point and C(x0,y0,z0) as projection centerThen P’ : x’=(x*z0-z*x0)/(z0-z), y’=(y*z0-z*y0)/(z0-z)For the perspective with 2 points: rotate around 1 axis + persp. roj. with 1 pointFor the perspective with 2 points: rotate around 2 axis + persp. proj. with 1 point
Ascending, descending
Use the vanishing points
Using the vanishing points
Parallel projectionsAssume object face of interest lies in principal plane, i.e., parallel to xy, yz, or zx planes. DOP = Direction of Projection, VPN = View Plane Normal1) Multiview Orthographic
VPN || a principal coordinate axisDOP || VPNshows single face, exact measurements
2) AxonometricVPN || a principal coordinate axisDOP || VPNadjacent faces, none exact, uniformly foreshortened (function of angle between face normal and DOP)
3) ObliqueVPN || a principal coordinate axisDOP || VPNadjacent faces, one exact, others uniformly
Multiview OrthographicUsed for:
engineering drawings of machines, machine partsworking architectural drawings
Pros:accurate measurement possibleall views are at same scale
Cons:does not provide “realistic” view or sense of 3D form
Usually need multiple views to get a three-dimensional feeling for object
Elevation – frontal view
Prop: maintain the distances and anglesDisadvantage: hard to understandFrontal view (projection on xOy plane):
such that x’=x, y’=y, z’=0
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Plan – top view
Projection on the plane xOz: Rx(-90o) + proj. on XOy:x’=x, y’=z, z’=0
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Profile – side view
Projection on the plane xOz: Rx(-90o) + proj. on XOy:x’=z, y’=y, z’=0
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Plan, elevation, profile
Front view: plane z=0, projection center at infinityon + halfaxisBack view: plane z=0, projection center at - infinityon - halfaxisLeft view: plane x=0, projection center at infinityon + halfaxisRight view: plane x=0, projection center at -infinityon - halfaxis
Plan, elevation, profile
Axonometric Projections
Same method as multiview orthographic projections, except projection plane not parallel to any of coordinate planes; parallel lines equally foreshortenedIsometric: Angles between all three principal axes equal (120º). Same scale ratio applies along each axisDimetric: Angles between two of the principal axes equal; need two scale ratiosTrimetric: Angles different between three principal axes; need three scale ratiosNote: different names for different views, but all part of a continuum of parallel projections of cube; these differ in where projection plane is relative to its cube
Isometric Projection
Construction of an isometric projection: projection plane cuts each principal axis by 45°
Axonometric projection
Obtained through 1-3 rotations around the axes + orthographic projection on xOyEx: OrtoxOyRx(u)Ry(v)
Isometric projection: v=arcsin(1/sqrt(3)),u=arcsin(1/2)
Isometric Projection
Used for:catalogue illustrationspatent office recordsfurniture designstructural design3d Modeling in real time (Maya, AutoCad, etc.)
Pros:don’t need multiple viewsillustrates 3D nature of objectmeasurements can be made to scale along principal axes
Cons:lack of foreshortening creates distorted appearancemore useful for rectangular than curved shapes
Still in use today when you want to see things in distance as well as things close up (e.g. strategy, simulation games)
Examples
SimCity IV StarCraft II
Dimetric
Parallel Proj.
Parallel proj.
Oblique Projections
Projectors at oblique angle to projection plane; view cameras have accordion housing, used for skyscrapersPros:
can present exact shape of one face of an object (can take accurate measurements): better for elliptical shapes than axonometric projections, better for “mechanical” viewinglack of perspective foreshortening makes comparison of sizes easierdisplays some of object’s 3D appearance
Cons:objects can look distorted if careful choice not made about position of projection plane (e.g., circles become ellipses)lack of foreshortening (not realistic looking)
Oblique projection
Main Types of Oblique Projections
Cavalier: Angle between projectors and projection plane is 45º. Perpendicular faces projected at full scale
Cabinet: Angle between projectors & projection plane: arctan(2) = 63.4º. Perpendicular faces projected at 50% scale
Examples
Projection plane parallel to circular face
Projection plane not parallel to circular face
Parallel projections
Different projections
Different projections
Different projections
Projective transformation – case 1
Projective transformation – case 1
x’/d=x/z, y’/d=y/z(X,Y,Z,W)=(x,y,z,z/d)(x’,y’,z’)=(X/W,Y/W,Z/W)=(xd/z,yd/z,d)
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Projective transformation – case 2
Projective transformation – case 2
x’/d=x/(z+d), y’/d=y/(z+d)(X,Y,Z,W)=(x,y,0,z/d+1)(x’,y’,z’)=(X/W,Y/W,Z/W)=(x/(z/d+1),y/(z/d+1),0)
1/d->0: orthographic projection on z=0
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Projective transformation – case 3
Projective transformation – case 3
Projective transformation – case 3
Projective transformation – case 3
Orthographic proj. on x=0: zp=0, Q=infinity, (dx,dy,dz)=(0,0,-1)Cavalier proj on z=0: zp=0, Q=infinity, (dx,dy,dz)=(cos(α),sin(α),-1)Cabinet proj. on z=0: zp=0, Q=infinity, (dx,dy,dz)=(1/2 cos(α),1/2 sin(α),-1)Perspective proj with 1 point: Q is finite, Mgeneral(0,0,1,0)T=(Q dx, Q dy, zp)
Description of visualization system
Projection plane: reference point Pr+ normal nReference viewing system: originie: Pr, 1 axis = n, 2 axis = v (image vertical, ex: y), 3 axis = u obtained from the product u x v = nProjection plane = u Pr vWindow in the projection plane: max/min coordinates in u Pr v + Cf (window center)Pd + projection indicator
Vector product
Example of viewing definition
Pr=(0,0,0) (xyz), n=(0,0,1) (xyz), v=(0,1,0) (xyz), Pd=(0.5,0.5,1) (uvn), Window=(0,1,0,1) (uvn) Projection: parallel
Example 2
Pr=(0,0,54), n=(0,0,1), v=(0,1,0), Pd=(8,6,30), Window (-1,17,-1,17), Projection: perspective
Example 3
Pr=(0,0,0), n=(0,0,1), v=(0,1,0), Pd=(8,8,100), Window=(-1,17,-1,17), Projection=parallel
Examples
Pr=(16,0,54), v=(0,1,0), Window=(-20,20,-5,35)
(a) n=(0,0,1) Pd=(20,25,20), perspective 1 point(b) n=(1,1,1), Pd=(0,0,10), parallel isometric(c) n=(1,0,1), Pd=(0,25,20sqrt(2), persp. 2 points
Vizualisation system
Viewing volume - analogy
Viewing volume
= part of WCS that is projected on the projection plane
1. Perspective projection : part of a pyramid with the peak in the projection center and the edges crossing through the window corner
2. Parallel projection: parallelepipedon with edges parallel with the projection direction
Viewing volumes
Viewing volume
Viewingvolume
Volum de vedere
Volum de vedere
Volum de vedere
Viewing volume
Viewing volumes
Canonical volumes
Canonical volume
Transforming the parallel volume in a canonical one
Transforming the perspective volume in a canonical one
Transforming the canonic volume from perspective proj to the one of parallel proj
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Can.view vol. persp->par.
3D visualization
Exemplu
