Numerical Computation of Parallel and Central Projections Numerical Computation of Parallel and Central Projections Szilvia B´ela and M´arta Szilv´asi-Nagy Department of Geometry, Institute of Mathematics Budapest University of Technology and Economics 3-7. September 2012, Rijeka, Croatia Sz. B´ela, M. Szilv´asi-Nagy Budapest University of Technology and Economics Numerical Computation of Parallel and Central Projections Motivation Aim of art and architecture: model the space in the plane such that it can be reconstructed from the image/images ↓ philosophers, artists, architectures and mathematicians were constructing such descriptive systems • axonometry – parallel projection keeps parallel lines • perspectivity – central projection gives the illusion of space from one point of view Sz. B´ela, M. Szilv´asi-Nagy Budapest University of Technology and Economics Numerical Computation of Parallel and Central Projections Coordinate systems World/Object coordinate system {O, x, y, z} - locate the points in 3d Descartes (orthogonal) coordinate system: • given origin – starting point of basis vectors • basis: orthogonal unit vectors {i, j, k} k j O i Sz. B´ela, M. Szilv´asi-Nagy Budapest University of Technology and Economics Numerical Computation of Parallel and Central Projections Coordinate systems World/Object coordinate system {O, x, y, z} - locate the points in 3d Descartes (orthogonal) coordinate system: • given origin – starting point of basis vectors • basis: orthogonal unit vectors {i, j, k} z −→ OP vector can be written as y k P the linear combination of i, j j and k −→ i OP = xi + yj + zk O x Coordinates of point P = (x, y, z) Sz. B´ela, M. Szilv´asi-Nagy Budapest University of Technology and Economics Numerical Computation of Parallel and Central Projections Coordinate systems World/Object coordinate system {O, x, y, z} - locate the points in 3d Image/Display coordinate system {O′,ξ,η,ζ} - locate the image of points • new origin – starting point of new basis vectors • basis: orthogonal unit vectors {e1, e2, e3} z y k j O i e3 x e2 O′ e1 Sz. B´ela, M. Szilv´asi-Nagy Budapest University of Technology and Economics Numerical Computation of Parallel and Central Projections Coordinate systems World/Object coordinate system {O, x, y, z} - locate the points in 3d Image/Display coordinate system {O′,ξ,η,ζ} - locate the image of points • new origin – starting point of new basis vectors • basis: orthogonal unit vectors {e1, e2, e3} z π: projection plane ζ η y • k the plane of the vectors j e1 and e2 • the plane, where we draw O i the 2d image of the e x 3 e object 2 π O′ e1 ξ Sz. B´ela, M. Szilv´asi-Nagy Budapest University of Technology and Economics Numerical Computation of Parallel and Central Projections Coordinate systems World/Object coordinate system {O, x, y, z} - locate the points in 3d Image/Display coordinate system {O′,ξ,η,ζ} - locate the image of points z ζ vP π: projection plane η y k P j vP : direction vector of projection in point P O i e3 x e2 π O′ e1 ξ Sz. B´ela, M. Szilv´asi-Nagy Budapest University of Technology and Economics Numerical Computation of Parallel and Central Projections Coordinate systems World/Object coordinate system {O, x, y, z} - locate the points in 3d Image/Display coordinate system {O′,ξ,η,ζ} - locate the image of points z v π: projection plane ζ P η y k P j vP : direction vector of projection in point P O i e3 x e2 P′ : the projection of point P P′ to the plane π in the direction v π of the vector P O′ e1 ξ Sz. B´ela, M. Szilv´asi-Nagy Budapest University of Technology and Economics Numerical Computation of Parallel and Central Projections Coordinate systems World/Object coordinate system {O, x, y, z} - locate the points in 3d Image/Display coordinate system {O′,ξ,η,ζ} - locate the image of points π: projection plane z ζ vP vP : direction vector of η y k P projection in point P j P : the projection of point P ′ O i e3 x −−→ e2 O′P′ = ξ′e1 + η′e2 +0e3 P′ π P′ = (ξ′, η′, 0) O′ e1 ξ Sz. B´ela, M. Szilv´asi-Nagy Budapest University of Technology and Economics Numerical Computation of Parallel and Central Projections Two Types of Projection Parallel Projection Central Projection (Axonometry) (Perspectivity) Sz. B´ela, M. Szilv´asi-Nagy Budapest University of Technology and Economics Numerical Computation of Parallel and Central Projections Two Types of Projection Parallel Projection (Axonometry) Sz. B´ela, M. Szilv´asi-Nagy Budapest University of Technology and Economics Numerical Computation of Parallel and Central Projections The Parallel Projection • the direction of the projection −−→ z v = OO′ y P ζ Q η O x v π O′ ξ Sz. B´ela, M. Szilv´asi-Nagy Budapest University of Technology and Economics Numerical Computation of Parallel and Central Projections The Parallel Projection • the direction of the projection −−→ z v = OO′ y P • the direction of projection is the same in each point P or Q ζ Q η O x ↓ v projection lines are parallel: −−→ −−→ P′ Q′ PP′kOO′ π O′ ξ Sz. B´ela, M. Szilv´asi-Nagy Budapest University of Technology and Economics Numerical Computation of Parallel and Central Projections Properties Keeps the parallelity: PQkRS → P′Q′kR′S ′ Therefore it keeps the ratio of the length of line segments along a line: |PQ| |P Q | = ′ ′ |PR| |P′R′| R ℓ Q P ℓ′ P′ Q′ R′ Sz. B´ela, M. Szilv´asi-Nagy Budapest University of Technology and Economics Numerical Computation of Parallel and Central Projections Special Case Oblique (general case) ζ η v • the direction of projection is arbitrary O′ ξ π Orthogonal (special case) • the direction of projection is ζ perpendicular to the projection η v plane • the length of a line segment decreases in the projection O′ ξ π Sz. B´ela, M. Szilv´asi-Nagy Budapest University of Technology and Economics Numerical Computation of Parallel and Central Projections Two Types of Projection Central Projection (Perspectivity) Sz. B´ela, M. Szilv´asi-Nagy Budapest University of Technology and Economics Numerical Computation of Parallel and Central Projections The Central Projection C • given a center point C along the line of OO′ z y P ζ Q η O x O′ ξ π Sz. B´ela, M. Szilv´asi-Nagy Budapest University of Technology and Economics Numerical Computation of Parallel and Central Projections The Central Projection C • given a center point C along the line of OO ′ z • the direction of projection y P passes through the point C ζ Q ↓ η O x direction of projection: −→ Q′ vP = CP P′ O′ ξ π Sz. B´ela, M. Szilv´asi-Nagy Budapest University of Technology and Economics Numerical Computation of Parallel and Central Projections Properties Keeps the cross ratio: [PQRS]=[P′Q′R′S ′] |PQ| |QS| |P Q | |Q S | = ′ ′ ′ ′ |RQ| |PS| |R′Q′| |P′S ′| ℓ S R Q P ℓ′ P′ Q′ R′ S ′ Sz. B´ela, M. Szilv´asi-Nagy Budapest University of Technology and Economics Numerical Computation of Parallel and Central Projections Monge’s System: orthogonal projections on mutually orthogonal image planes P′′ π2 P P′′ Q′′ Q Q′′ Q′ Q′ P′ P′ π1 The two image planes, π1 and π2 are usually the xy and yz coordinate planes of the world coordinate system. The projection of a point is constructed with the help of the projection of its coordinate lines in the world coordinate system. Sz. B´ela, M. Szilv´asi-Nagy Budapest University of Technology and Economics Numerical Computation of Parallel and Central Projections Projection Method • 1. step: Basis transformation • 2. step: Projection Sz. B´ela, M. Szilv´asi-Nagy Budapest University of Technology and Economics Numerical Computation of Parallel and Central Projections Step 1: Basis Transformation Projections are computed in the image coordinate system. Aim: find the coordinates of the point P given in the world coordinate system {O, x, y, z} in the image coordinate system {O′,ξ,η,ζ} • translation of the world coordinate system {O, x, y, z} O → O′ −−→ compute O′P in the world coordinate system • transformation of basis vectors {i, j, k} → {e1, e2, e3} −−→ compute the coordinates of O′P in the image coordinate system Sz. B´ela, M. Szilv´asi-Nagy Budapest University of Technology and Economics Numerical Computation of Parallel and Central Projections • translation −−→ −→ −−→ O′P = OP − OO′ • transformation If the world coordinates of Pworld = (x, y, z) in the translated system, then −−→ O′P = xi + yj + zk so we have to find the coordinates of the basis vectors i, j and k in the new basis {e1, e2, e3}: i = b11e1 + b12e2 + b13e3 j = b21e1 + b22e2 + b23e3 k = b31e1 + b32e2 + b33e3. j=1,2,3 The basis transformation matrix: B = {bij }i=1,2,3 New coordinates in the image coordinate system: T Pimg = (ξ,η,ζ)= B · (x, y, z) . Sz. B´ela, M. Szilv´asi-Nagy Budapest University of Technology and Economics Numerical Computation of Parallel and Central Projections Example The image coordinate system lies in the diagonal plane of the prism defined by (0, 0, 0) and (1, 2, 2) World coord.sys.: O(0, 0, 0) Image coord.sys.: O′(0.5, 1, 0) i(1, 0, 0) e1(1/√5, 2/√5, 0) j(0, 1, 0) e2(0, 0, 1) k(0, 0, 1) e (2/√5, 1/√5, 0) 3 − z 1 2 ξ 0 x 0.5 √5 √5 − D η = 0 0 1 y 1 k x − ζ 2 1 0 z A √5 − √5 x Cx e2 Bx x 1 1 1 1 0 0 0 0 O j y 0 0 2 2 2 2 0 0 A D z 0 1 0 1 0 1 0 1 e i 1 A Ax B Bx C Cx D Dx O′ C x e3 y 3 3 5 5 3 3 5 5 ξ − − − − B 2√5 2√5 2√5 2√5 2√5 2√5 2√5 2√5 η 0 1 0 1 0 1 0 1 2 2 2 2 ζ 0 0 − − 0 0 √5 √5 √5 √5 Sz.