3D graphics eBook - Course Materials Repository
3D graphics eBook - Course Materials Repository 3D graphics eBook - Course Materials Repository
Isosurface 67 Examples of isosurfaces are 'Metaballs' or 'blobby objects' used in 3D visualisation. A more general way to construct an isosurface is to use the function representation and the HyperFun language. External links • Isosurface Polygonization [1] References [1] http:/ / www2. imm. dtu. dk/ ~jab/ gallery/ polygonization. html Lambert's cosine law Isosurface of vorticity trailed from a propeller In optics, Lambert's cosine law says that the radiant intensity observed from an ideal diffusely reflecting surface (a Lambertian surface) is directly proportional to the cosine of the angle θ between the observer's line of sight and the surface normal. The law is also known as the cosine emission law or Lambert's emission law. It is named after Johann Heinrich Lambert, from his Photometria, published in 1760. An important consequence of Lambert's cosine law is that when a Lambertian surface is viewed from any angle, it has the same apparent radiance. This means, for example, that to the human eye it has the same apparent brightness (or luminance). It has the same radiance because, although the emitted power from a given area element is reduced by the cosine of the emission angle, the apparent size (solid angle) of the observed area, as seen by a viewer, is decreased by a corresponding amount. Therefore, its radiance (power per unit solid angle per unit projected source area) is the same. For example, in the visible spectrum, the Sun is not a Lambertian radiator; its brightness is a maximum at the center of the solar disk, an example of limb darkening. A black body is a perfect Lambertian radiator. Lambertian scatterers When an area element is radiating as a result of being illuminated by an external source, the irradiance (energy or photons/time/area) landing on that area element will be proportional to the cosine of the angle between the illuminating source and the normal. A Lambertian scatterer will then scatter this light according to the same cosine law as a Lambertian emitter. This means that although the radiance of the surface depends on the angle from the normal to the illuminating source, it will not depend on the angle from the normal to the observer. For example, if the moon were a Lambertian scatterer, one would expect to see its scattered brightness appreciably diminish towards the terminator due to the increased angle at which sunlight hit the surface. The fact that it does not diminish illustrates that the moon is not a Lambertian scatterer, and in fact tends to scatter more light into the oblique angles blade
Lambert's cosine law 68 than would a Lambertian scatterer. Details of equal brightness effect The situation for a Lambertian surface (emitting or scattering) is illustrated in Figures 1 and 2. For conceptual clarity we will think in terms of photons rather than energy or luminous energy. The wedges in the circle each represent an equal angle dΩ, and for a Lambertian surface, the number of photons per second emitted into each wedge is proportional to the area of the wedge. It can be seen that the length of each wedge is the product of the diameter of the circle and cos(θ). It can also be seen that the maximum rate of photon emission per unit solid angle is along the normal and diminishes to zero for θ = 90°. In mathematical terms, the radiance along the normal is I photons/(s·cm 2 ·sr) and the number of photons per second emitted into the vertical wedge is I dΩ dA. The number of photons per second emitted into the wedge at angle θ is I cos(θ) dΩ dA. Figure 2 represents what an observer sees. The observer directly above the area element will be seeing the scene through an aperture of area dA 0 and the area element dA will subtend a (solid) angle of dΩ 0 . We can assume without loss of generality that the aperture happens to subtend solid angle dΩ when "viewed" from the emitting area element. This normal observer will then be recording I dΩ dA photons per second and so will be measuring a radiance of photons/(s·cm 2 ·sr). Figure 1: Emission rate (photons/s) in a normal and off-normal direction. The number of photons/sec directed into any wedge is proportional to the area of the wedge. Figure 2: Observed intensity (photons/(s·cm 2 ·sr)) for a normal and off-normal observer; dA 0 is the area of the observing aperture and dΩ is the solid angle subtended by the aperture from the viewpoint of the emitting area element. The observer at angle θ to the normal will be seeing the scene through the same aperture of area dA 0 and the area element dA will subtend a (solid) angle of dΩ 0 cos(θ). This observer will be recording I cos(θ) dΩ dA photons per second, and so will be measuring a radiance of which is the same as the normal observer. photons/(s·cm 2 ·sr),
- Page 21 and 22: Binary space partitioning 16 togeth
- Page 23 and 24: Binary space partitioning 18 Other
- Page 25 and 26: Binary space partitioning 20 [1] Bi
- Page 27 and 28: Bounding interval hierarchy 22 Prop
- Page 29 and 30: Bounding volume 24 bounding boxes b
- Page 31 and 32: Bump mapping 26 Bump mapping Bump m
- Page 33 and 34: CatmullClark subdivision surface 28
- Page 35 and 36: CatmullClark subdivision surface 30
- Page 37 and 38: Conversion between quaternions and
- Page 39 and 40: Cube mapping 34 Advantages Cube map
- Page 41 and 42: Cube mapping 36 Related A large set
- Page 43 and 44: Diffuse reflection 38 2), or, of co
- Page 45 and 46: Displacement mapping 40 Meaning of
- Page 47 and 48: DooSabin subdivision surface 42 Ext
- Page 49 and 50: False radiosity 44 False radiosity
- Page 51 and 52: Geometry pipelines 46 Geometry pipe
- Page 53 and 54: Global illumination 48 Rendering wi
- Page 55 and 56: Gouraud shading 50 Gouraud shading
- Page 57 and 58: Graphics pipeline 52 Graphics pipel
- Page 59 and 60: Graphics pipeline 54 References 1.
- Page 61 and 62: Hidden surface determination 56 imp
- Page 63 and 64: High dynamic range rendering 58 Hig
- Page 65 and 66: High dynamic range rendering 60 Ton
- Page 67 and 68: High dynamic range rendering 62 Fro
- Page 69 and 70: High dynamic range rendering 64 •
- Page 71: Irregular Z-buffer 66 Applications
- Page 75 and 76: Lambertian reflectance 70 Lambertia
- Page 77 and 78: Level of detail 72 Well known appro
- Page 79 and 80: Level of detail 74 Hierarchical LOD
- Page 81 and 82: Newell's algorithm 76 Newell's algo
- Page 83 and 84: Non-uniform rational B-spline 78 Us
- Page 85 and 86: Non-uniform rational B-spline 80 of
- Page 87 and 88: Non-uniform rational B-spline 82 ar
- Page 89 and 90: Non-uniform rational B-spline 84 Ex
- Page 91 and 92: Normal mapping 86 How it works To c
- Page 93 and 94: OrenNayar reflectance model 88 Oren
- Page 95 and 96: OrenNayar reflectance model 90 , ,
- Page 97 and 98: Painter's algorithm 92 The algorith
- Page 99 and 100: Parallax mapping 94 • Parallax Ma
- Page 101 and 102: Particle system 96 A cube emitting
- Page 103 and 104: Path tracing 98 History Further inf
- Page 105 and 106: Path tracing 100 Scattering distrib
- Page 107 and 108: Phong reflection model 102 Visual i
- Page 109 and 110: Phong reflection model 104 Because
- Page 111 and 112: Phong shading 106 Visual illustrati
- Page 113 and 114: Photon mapping 108 Rendering (2nd p
- Page 115 and 116: Photon tracing 110 Advantages and d
- Page 117 and 118: Potentially visible set 112 • Can
- Page 119 and 120: Potentially visible set 114 Externa
- Page 121 and 122: Procedural generation 116 increases
Isosurface 67<br />
Examples of isosurfaces are 'Metaballs' or 'blobby objects' used in <strong>3D</strong><br />
visualisation. A more general way to construct an isosurface is to use<br />
the function representation and the HyperFun language.<br />
External links<br />
• Isosurface Polygonization [1]<br />
References<br />
[1] http:/ / www2. imm. dtu. dk/ ~jab/ gallery/ polygonization. html<br />
Lambert's cosine law<br />
Isosurface of vorticity trailed from a propeller<br />
In optics, Lambert's cosine law says that the radiant intensity observed from an ideal diffusely reflecting surface (a<br />
Lambertian surface) is directly proportional to the cosine of the angle θ between the observer's line of sight and the<br />
surface normal. The law is also known as the cosine emission law or Lambert's emission law. It is named after<br />
Johann Heinrich Lambert, from his Photometria, published in 1760.<br />
An important consequence of Lambert's cosine law is that when a Lambertian surface is viewed from any angle, it<br />
has the same apparent radiance. This means, for example, that to the human eye it has the same apparent brightness<br />
(or luminance). It has the same radiance because, although the emitted power from a given area element is reduced<br />
by the cosine of the emission angle, the apparent size (solid angle) of the observed area, as seen by a viewer, is<br />
decreased by a corresponding amount. Therefore, its radiance (power per unit solid angle per unit projected source<br />
area) is the same. For example, in the visible spectrum, the Sun is not a Lambertian radiator; its brightness is a<br />
maximum at the center of the solar disk, an example of limb darkening. A black body is a perfect Lambertian<br />
radiator.<br />
Lambertian scatterers<br />
When an area element is radiating as a result of being illuminated by an external source, the irradiance (energy or<br />
photons/time/area) landing on that area element will be proportional to the cosine of the angle between the<br />
illuminating source and the normal. A Lambertian scatterer will then scatter this light according to the same cosine<br />
law as a Lambertian emitter. This means that although the radiance of the surface depends on the angle from the<br />
normal to the illuminating source, it will not depend on the angle from the normal to the observer. For example, if<br />
the moon were a Lambertian scatterer, one would expect to see its scattered brightness appreciably diminish towards<br />
the terminator due to the increased angle at which sunlight hit the surface. The fact that it does not diminish<br />
illustrates that the moon is not a Lambertian scatterer, and in fact tends to scatter more light into the oblique angles<br />
blade