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Three dimensional transformations
- 2. 3D transformation methods are extended from 2D methods by including considerations for the z coordinate A 3D homogenous coordinate is represented as a four-element column vector ◦ Each geometric transformation operator is a 4 by 4 matrix
- 3. Translation of a Point zyx tzztyytxx ',',' 11000 100 010 001 1 ' ' ' z y x t t t z y x z y x P’=P·T (x,y,z) T=(tx,ty,tz) (x’,y’,z’) x z y
- 4. Uniform Scaling (Scaling relative to the coordinate origin) zyx szzsyysxx ',',' 11000 000 000 000 1 ' ' ' z y x s s s z y x z y x x z y P’ = S·P
- 5. Scaling with a Selected fixed point (xf, yf , zf) 11000 100 010 001 1000 000 000 000 1000 100 010 001 1 ' ' ' ),,(),,(),,( z y x z y x s s s z y x z y x zyxTsssSzyxT f f f z y x f f f fffzyxfff x x x xzzzz y y y y Original position Translate Scaling Inverse Translate
- 6. Positive rotation angles produce counterclockwise rotations about a coordinate axis, assuming that we are looking in the negative direction along that coordinate axis Coordinate-Axis Rotations ◦ Z-axis rotation ◦ X-axis rotation ◦ Y-axis rotation General 3D Rotations ◦ Rotation about an axis that is parallel to one of the coordinate axes ◦ Rotation about an arbitrary axis
- 7. Z-Axis Rotation X-Axis Rotation Y-Axis Rotation 11000 0100 00cossin 00sincos 1 ' ' ' z y x z y x 11000 0cossin0 0sincos0 0001 1 ' ' ' z y x z y x 11000 0cos0sin 0010 0sin0cos 1 ' ' ' z y x z y x
- 8. z-axis rotation ◦ x’=x cos θ - y sin θ ◦ y’=x sin θ + y cos θ ◦ z’=z or, P’ = Rz(θ)·P 11000 0100 00cossin 00sincos 1 ' ' ' z y x z y x
- 9. z-axis rotation ◦ x’=x cos θ - y sin θ ◦ y’=x sin θ + y cos θ ◦ z’=z Other axis rotations ◦ x→ y→ z→ x x-axis rotation ◦ y’=y cos θ - z sin θ ◦ z’=y sin θ + z cos θ ◦ x’=x y-axis rotation ◦ z’=z cos θ - x sin θ ◦ x’=z sin θ + x cos θ ◦ y’=y or, P’ = Rx(θ)·P or, P’ = Ry(θ)·P
- 10. General 3D Rotations : CASE 1 • Rotation about an Axis that is Parallel to One of the Coordinate Axes – Translate the object so that the rotation axis coincides with the parallel coordinate axis – Perform the specified rotation about that axis – Translate the object so that the rotation axis is moved back to its original position – Any coordinate position P on the object in this fig. is transformed with the sequence shown as below P’ = T-1·Rx(θ)·T·P
- 11. Rotation about an Arbitrary Axis Basic Idea 1. Translate (x1, y1, z1) to the origin 2. Rotate (x’2, y’2, z’2) on to the z axis 3. Rotate the object around the z-axis 4. Rotate the axis to the original orientation 5. Translate the rotation axis to the original position (x2,y2,z2) (x1,y1,z1) x z y R-1 T-1 R T TRRRRRTR xyzyx 111
- 12. Step 1. Translation 1000 100 010 001 1 1 1 z y x T (x2,y2,z2) (x1,y1,z1) x z y
- 13. Step 2. Establish [ TR ] x x axis 1000 0//0 0//0 0001 1000 0cossin0 0sincos0 0001 dcdb dbdc x R (a,b,c) (0,b,c) Projected Point Rotated Point d c cb c d b cb b 22 22 cos sin x y z
- 14. cgvr.korea.ac.kr Step 3. Rotate about y axis by (a,b,c) (a,0,d) l d 22 222222 cos,sin cbd dacbal l d l a 1000 0/0/ 0010 0/0/ 1000 0cos0sin 0010 0sin0cos ldla lald y Rx y Projected Point z Rotated Point
- 15. Step 4. Rotate about z axis by the desired angle l 1000 0100 00cossin 00sincos zR y x z
- 16. Step 5. Apply the reverse transformation to place the axis back in its initial position x y l l z 1000 0cos0sin 0010 0sin0cos 1000 0cossin0 0sincos0 0001 1000 100 010 001 1 1 1 111 z y x yx RRT TRRRRRTR xyzyx 111
- 17. Find the new coordinates of a unit cube 90º-rotated about an axis defined by its endpoints A(2,1,0) and B(3,3,1). A Unit Cube
- 18. Step1. Translate point A to the origin A’(0,0,0) x z y B’(1,2,1) 1000 0100 1010 2001 T
- 19. x z y l 1000 0 5 5 5 52 0 0 5 52 5 5 0 0001 xR 6121 5 5 5 1 cos 5 52 5 2 12 2 sin 222 22 l B’(1,2,1) Projected point (0,2,1) B”(1,0,5) Step 2. Rotate axis A’B’ about the x axis by and angle , until it lies on the xz plane.
- 20. x z y l 1000 0 6 30 0 6 6 0010 0 6 6 0 6 30 yR 6 30 6 5 cos 6 6 6 1 sin B”(1,0, 5) (0,0,6) Step 3. Rotate axis A’B’’ about the y axis by and angle , until it coincides with the z axis.
- 21. Step 4. Rotate the cube 90° about the z axis ◦ Finally, the concatenated rotation matrix about the arbitrary axis AB becomes, TRRRRRTR xyzyx 90111 1000 0100 0001 0010 90zR
- 23. PRP 11111111 076.0091.0560.0726.0817.0650.0301.1467.1 483.0409.0151.1225.1184.0258.0484.0558.0 891.2909.1742.1725.2816.2834.1667.1650.2 11111111 10011001 00001111 11001100 1000 560.0167.0741.0650.0 151.1075.0667.0742.0 742.1983.0075.0166.0 P Multiplying R(θ) by the point matrix of the original cube
- 24. A 3-D Reflection can be performed relative to a selected reflection axis or w.r.t selected reflection plane. The 3-D reflection matrixes are set up similarly to those for 2-D. In 2-D , Reflection w.r.t axis is equivalent to 180 degree rotations about the axis in 3- D space whereas ,in 3-D Reflection w.r.t a plane are equivalent to 180 degree rotations in 4-D space. 3D Transformation 24
- 25. Other Transformations : REFLECTION Reflection Relative to the XY Plane x z y x z y 11000 0100 0010 0001 1 ' ' ' z y x z y x Reflection Relative to the XZ Plane 11000 0100 0010 0001 1 ' ' ' z y x z y x x z y x z y 11000 0100 0010 0001 1 ' ' ' z y x z y x Reflection Relative to the YZ Plane x z y z y x
- 26. Other Transformations : SHEARING • Shearing transformation are used to modify the shape of the object and they are useful in 3-D viewing for obtaining General Projection transformations. • Z-axis 3-D Shear transformation • The effect of this transformation matrix is to alter the x and y co-ordinate values by an amount that is proportional to the z-value, while leaving z co-ordinate unchanged. Boundaries of the plane that are perpendicular to z-axis are thus shifted proportional to z-value. 11000 0100 010 001 1 ' ' ' z y x b a z y x
- 27. X-axis 3-D Shear transformation Y-axis 3-D Shear transformation 11000 010 0010 001 1 ' ' ' z y x b a z y x 11000 010 001 0001 1 ' ' ' z y x b a z y x
- 28. 3D Transformation Slide 28 Viewing in 3D
- 29. • Man-made objects often have “cube-like” shape. These objects have 3 principle axis. 3D Transformation Slide 29
- 30. 3D Transformation Slide 30 • How do we map 3D objects to 2D space? Display device (a screen) is 2D… • 2D window to world.. and a viewport on the 2D surface. • Clip what won't be shown in the 2D window, and map the remainder to the viewport. 2D to 2D is straight forward… • Solution : Transform 3D objects on to a 2D plane using projections 3D to 2D is more complicated…
- 31. • In 3D… – View volume in the world – Projection onto the 2D projection plane – A viewport to the view surface • Process… – 1… clip against the view volume, – 2… project to 2D plane, or window, – 3… map to viewport. 3D Transformation Slide 31
- 32. • Conceptual Model of the 3D viewing process 3D Transformation 32
- 33. 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 333D Transformation
- 34. • 2 types of projections – PERSPECTIVE and PARALLEL. • Key factor is the center of projection. – if distance to center of projection is finite : PERSPECTIVE – if distance to center of projection is infinite : PARALLEL 3D Transformation Slide 34
- 35. 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) In parallel projection, coordinate positions are transformed to the view plane along parallel lines. 353D Transformation
- 36. • Perspective projection + Size varies inversely with distance - looks realistic – Distance and angles are not (in general) preserved – Parallel lines do not (in general) remain parallel • Parallel projection + Good for exact measurements + Parallel lines remain parallel – Angles are not (in general) preserved – Less realistic looking
- 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 383D Transformation
- 39. • 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 principle vanishing point. – There are at most 3 such points, corresponding to the number of axes cut by the projection plane. 393D Transformation
- 40. • Certain set of parallel lines appear to meet at a different point – The Vanishing point for this direction • Principle 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 Vanishing points 403D Transformation
- 41. Classes of Perspective Projection • One-Point Perspective • Two-Point Perspective • Three-Point Perspective 413D Transformation
- 43. Two-point perspective projection: 433D Transformation
- 44. 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 443D Transformation
- 45. Affine Transformations • Affine transformations are combinations of … – Linear transformations, and – Translations • Properties of affine transformations: – Origin does not necessarily map to origin – Lines map to lines – Parallel lines remain parallel – Ratios are preserved – Closed under composition w y x fed cba w y x 100 ' '
- 46. Perspective Transformations • Projective transformations … – Affine transformations, and – Projective warps • Properties of projective transformations: – Origin does not necessarily map to origin – Lines map to lines – Parallel lines do not necessarily remain parallel – Ratios are not preserved – Closed under composition w y x ihg fed cba w y x ' ' '
- 52. Center of projection is at infinity ◦ Direction of projection (DOP) same for all points DOP View Plane
- 53. • 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. Parallel Projections 533D Transformation
- 54. when the projection is perpendicular to the view plane when the projection is not perpendicular to the view plane • Orthographic projection Oblique projection 543D Transformation
- 55. – 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. Orthographic (or orthogonal) projections: 553D Transformation
- 57. 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. 573D Transformation
- 58. OBLIQUE PARALLEL PROJECTIONS : 583D Transformation
- 59. Cavalier projection: • All lines perpendicular to the projection plane are projected with no change in length. OBLIQUE PARALLEL PROJECTIONS : Cavalier and Cabinet 593D Transformation
- 60. Oblique Projections : CAVALIER PROJECTION • DOP not perpendicular to view plane Cavalier (DOP = 45 o ) tan() = 1
- 61. 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. 613D Transformation Oblique Projections : CABINET PROJECTION
- 62. Oblique Projections : CABINET PROJECTION H&B • DOP not perpendicular to view plane Cabinet (DOP = 63.4 o ) tan() = 2
- 63. 633D Transformation Remaining Topics - (REFER CLASS NOTES) Transformation Matrix for Oblique Projection of a 3-D point. General Projection Transformations General Parallel Projection Transformation General Perspective Projection Transformation View Volumes for Projections