computer graphics iii spherical integrals, light...
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Computer Graphics III Spherical integrals, Light & Radiometry – Exercises
Jaroslav Křivánek, MFF UK
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Reminders & org
◼ Renderings due next week
❑ Upload to google drive, show on the big screen, 5 minutes per team (how many teams do we have)
◼ Papers for presentations in the lab – 7.11., 21.11,
❑ ACM TOG special issue on production renderinghttps://dl.acm.org/citation.cfm?id=3243123&picked=prox
◼ Reminder – choose papers for the exam
❑ http://kesen.realtimerendering.com/
◼ Log your choices here
❑ https://docs.google.com/document/d/128e4Dgh0IvH64DI6Ohu2eRGth0m5i8WlKpDwNyJzpVM/edit?usp=sharing
◼ Decide assignments track vs. individual project track by Wed, Oct 31st 2018.
CG III (NPGR010) - J. Křivánek
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PEN & PAPER EXERCISES
CG III (NPGR010) - J. Křivánek
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◼ Calculate the surface area of a unit sphere.
◼ Calculate the surface area of a spherical cap delimited by the angle q0 measured from the north pole.
◼ Calculate the surface area of a spherical wedge with angle f0.
Surface area of a (subset of a) sphere
CG III (NPGR010) - J. Křivánek
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◼ What is the solid angle under which we observe an (infinite) plane from a point outside of the plane?
◼ Calculate the solid angle under which we observe a sphere with radius R, the center of which is at the distance D from the observer.
Solid angle
CG III (NPGR010) - J. Křivánek
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Isotropic point light
◼ Q: What is the emitted power (flux) of an isotropic point light source with intensity that is a constant I in all directions?
CG III (NPGR010) - J. Křivánek
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Isotropic point light
◼ A: Total flux:
I
I
I
substituteI
q
q
4
cos2
ddsin
ddsind
:d)(
0
2
0 0
=
−=
=
===
= =
4
=I
CG III (NPGR010) - J. Křivánek
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Cosine spot light
◼ What is the power (flux) of a point source with radiant intensity given by:
sdII },0max{)( 0
=
CG III (NPGR010) - J. Křivánek
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◼ What is the power (flux) of a point light source with radiant intensity given by:
Spotlight with linear angular fall-off
CG III (NPGR010) - J. Křivánek
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Výpočet
CG III (NPGR010) - J. Křivánek
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◼ What is the irradiance E(x) at point x due to a uniform Lambertian area source observed from point x under the solid angle ?
Irradiance due to a Lambertian light source
CG III (NPGR010) - J. Křivánek
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CG III (NPGR010) - J. Křivánek
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CG III (NPGR010) - J. Křivánek
Based in these hints, calculate the solid angle under which we observe the Sun. (We assume the Sun is at the zenith.)
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◼ What is the irradiance at point x on a plane due to a point source with intensity I() placed at the height habove the plane.
◼ The segment connecting point xto the light position p makes the angle q with the normal of the plane.
Irradiance due to a point source
CG III (NPGR010) - J. Křivánek
dAx
p
d
q
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Irradiance due to a point source
◼ Irradiance of a point on a plane lit by a point source:
2
3
2
cos)(
cos)(
)(
)()(
hI
I
dA
dI
dA
dE
q
q
xp
xpxp
xp
xx
→=
−→=
→=
=
dAx
p
d
q
CG III (NPGR010) - J. Křivánek
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Area light sources
◼ Emission of an area light source is fully described by the emitted radiance Le(x,) for all positions on the source xand all directions .
◼ The total emitted power (flux) is given by an integral of Le(x,) over the surface of the light source and all directions.
ALA H
e ddcos),()(
= qx
x
CG III (NPGR010) - J. Křivánek
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◼ What is the relationship between the emitted radiant exitance (radiosity) Be(x) and emitted radiance Le(x, ) for a Lambertian area light source?
Lambertian source =
emitted radiance does not depend on the direction
Le(x, ) = Le(x).
Diffuse (Lambertian) light source
CG III (NPGR010) - J. Křivánek
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Diffuse (Lambertian) light source
◼ Le(x, ) is constant in
◼ Radiosity: Be(x) = Le(x)
)(
dcos)(
dcos),()(
)(
)(
x
x
xx
x
x
e
He
Hee
L
L
LB
q
q
=
=
=
CG III (NPGR010) - J. Křivánek
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◼ What is the total emitted power (flux) of a uniformLambertian area light source which emits radiance Le
❑ Uniform source – radiance does not depend on the position, Le(x, ) = Le = const.
Uniform Lambertian light source
CG III (NPGR010) - J. Křivánek
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Uniform Lambertian light source
◼ Le(x, ) is constant in x and
e = A Be = A Le
CG III (NPGR010) - J. Křivánek