Lesson 4 – Math / Lighting
PV227 – GPU Rendering
Jiˇrí Chmelík, Jan ˇCejka
Fakulta informatiky Masarykovy univerzity
6. 10. 2015
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 1 / 37
Outline
1 Basics – Repetition
2 Issues with Normals
3 Lighting
Light Components: Ambient
Light-source type: Directional Light
Shading types: Gouraud Shading
Light Components: Diffuse
Shading types: Flat Shading
Light Components: Ambient plus Diffuse
Light Components: Specular
Light Components: Ambient plus Diffuse plus Specular
Shading types: Phong Shading
Light-source type: Point Light
Light-source type: Spot Light
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 2 / 37
Points and Vectors
points (before projection) are quadruples: (x, y, z, 1.0),
can be transformed with a 4 × 4 matrix,
vectors are also quadruples: (x, y, z, 0.0),
can be transformed with a 4 × 4 or 3 × 3 matrix.
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 3 / 37
Transformation
points are transformed to eye space with modelview matrix,
vectors constructed from points (e.g. P2 − P1) are also
transformed with this matrix,
normals are not!
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 4 / 37
Outline
1 Basics – Repetition
2 Issues with Normals
3 Lighting
Light Components: Ambient
Light-source type: Directional Light
Shading types: Gouraud Shading
Light Components: Diffuse
Shading types: Flat Shading
Light Components: Ambient plus Diffuse
Light Components: Specular
Light Components: Ambient plus Diffuse plus Specular
Shading types: Phong Shading
Light-source type: Point Light
Light-source type: Spot Light
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 5 / 37
Normal Transformation Error
Figure: Taken from lighthouse3d.com
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 6 / 37
Normal Transformation Solution
caused by non-uniform scale,
M is the modelview matrix, t is tangent vector (P2 − P1) and I is
identity,
we need another matrix (N) for transforming normal n.
(M × t) • (N × n) = 0
(M × t)T
× (N × n) = 0
tT
× MT
× N × n = 0
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 7 / 37
Normal Transformation Solution (cont.)
tT
× MT
× N × n = 0
t • n = 0 ⇒ tT × n = 0 ⇒ MT × N = I
MT
× N = I
(MT
)−1
× MT
× N = (MT
)−1
N = (MT
)−1
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 8 / 37
Normal Transformation Result
N is inverse transpose of M (3 × 3 submatrix of M),
for orthogonal matrices: AT = A−1 (rotation is orthogonal),
if M is orthogonal, then: M = (MT )−1 ⇒ N = M.
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 9 / 37
Renormalization
normals must be of unit length,
can be destroyed by normal transformation → must be normalized
in vertex shader, after the transformation,
interpolation can also destroy vector length → must be normalized
in fragment shader, before we do anything with it.
Figure: Taken from lighthouse3d.com
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 10 / 37
Outline
1 Basics – Repetition
2 Issues with Normals
3 Lighting
Light Components: Ambient
Light-source type: Directional Light
Shading types: Gouraud Shading
Light Components: Diffuse
Shading types: Flat Shading
Light Components: Ambient plus Diffuse
Light Components: Specular
Light Components: Ambient plus Diffuse plus Specular
Shading types: Phong Shading
Light-source type: Point Light
Light-source type: Spot Light
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 11 / 37
Lighting
Computation of light’s interaction with surfaces,
in most cases – huge cheat; major simplification otherwise.
Light:
light components: ambient, diffuse and specular,
shading types: flat, gouraud and phong,
light-source types: directional, point and spot light,
shadows... no shadow, no bouncing of light.
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 12 / 37
Outline
1 Basics – Repetition
2 Issues with Normals
3 Lighting
Light Components: Ambient
Light-source type: Directional Light
Shading types: Gouraud Shading
Light Components: Diffuse
Shading types: Flat Shading
Light Components: Ambient plus Diffuse
Light Components: Specular
Light Components: Ambient plus Diffuse plus Specular
Shading types: Phong Shading
Light-source type: Point Light
Light-source type: Spot Light
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 13 / 37
Ambient Lighting
approximates lighting after infinite number of bounces,
homogeneous,
prevents black areas that look unnatural,
usually chosen as fraction of the diffuse (material) color,
I = Ka.
Figure: Ambient spheres
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 14 / 37
Outline
1 Basics – Repetition
2 Issues with Normals
3 Lighting
Light Components: Ambient
Light-source type: Directional Light
Shading types: Gouraud Shading
Light Components: Diffuse
Shading types: Flat Shading
Light Components: Ambient plus Diffuse
Light Components: Specular
Light Components: Ambient plus Diffuse plus Specular
Shading types: Phong Shading
Light-source type: Point Light
Light-source type: Spot Light
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 15 / 37
Light-source type: Directional Light
Figure: Taken from lighthouse3d.com
far away light,
defined by a direction vector (position is irelevant),
can represent e.g. the sun.
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 16 / 37
Outline
1 Basics – Repetition
2 Issues with Normals
3 Lighting
Light Components: Ambient
Light-source type: Directional Light
Shading types: Gouraud Shading
Light Components: Diffuse
Shading types: Flat Shading
Light Components: Ambient plus Diffuse
Light Components: Specular
Light Components: Ambient plus Diffuse plus Specular
Shading types: Phong Shading
Light-source type: Point Light
Light-source type: Spot Light
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 17 / 37
Shading types: Gouraud Shading
per vertex shading,
interpolation of vertex colors,
unable to capture lighting details inside polygons.
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 18 / 37
Outline
1 Basics – Repetition
2 Issues with Normals
3 Lighting
Light Components: Ambient
Light-source type: Directional Light
Shading types: Gouraud Shading
Light Components: Diffuse
Shading types: Flat Shading
Light Components: Ambient plus Diffuse
Light Components: Specular
Light Components: Ambient plus Diffuse plus Specular
Shading types: Phong Shading
Light-source type: Point Light
Light-source type: Spot Light
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 19 / 37
Light Components: Diffuse
simulate light’s interaction with perfectly diffuse material,
light angle dependent,
significant color component,
I = cos(α) · Kd .
Figure: Diffuse spheres
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 20 / 37
Light Components: Diffuse (cont.)
Figure: Taken from lighthouse3d.com
amount of incoming light diminishes with increasing angle,
cos(α) = L•N
|L|·|N|
,
normalized vectors: I = (L • N) · Kd ,
all vectors must be in same space (usually defined in
world space, computation in camera space).
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 21 / 37
Outline
1 Basics – Repetition
2 Issues with Normals
3 Lighting
Light Components: Ambient
Light-source type: Directional Light
Shading types: Gouraud Shading
Light Components: Diffuse
Shading types: Flat Shading
Light Components: Ambient plus Diffuse
Light Components: Specular
Light Components: Ambient plus Diffuse plus Specular
Shading types: Phong Shading
Light-source type: Point Light
Light-source type: Spot Light
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 22 / 37
Flat Shading
per primitive shading,
no interpolation,
unable to capture smooth changes in light intensity.
Figure: Flat shading
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 23 / 37
Outline
1 Basics – Repetition
2 Issues with Normals
3 Lighting
Light Components: Ambient
Light-source type: Directional Light
Shading types: Gouraud Shading
Light Components: Diffuse
Shading types: Flat Shading
Light Components: Ambient plus Diffuse
Light Components: Specular
Light Components: Ambient plus Diffuse plus Specular
Shading types: Phong Shading
Light-source type: Point Light
Light-source type: Spot Light
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 24 / 37
Light Components: Ambient plus Diffuse
light from various sources can be combined (added),
combination of ambient and diffuse prevents black areas,
I = Ka + cos(α) · Kd ,
value should not be outside the [0.0, 1.0] range.
Figure: Ambient + Diffuse spheres
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 25 / 37
Outline
1 Basics – Repetition
2 Issues with Normals
3 Lighting
Light Components: Ambient
Light-source type: Directional Light
Shading types: Gouraud Shading
Light Components: Diffuse
Shading types: Flat Shading
Light Components: Ambient plus Diffuse
Light Components: Specular
Light Components: Ambient plus Diffuse plus Specular
Shading types: Phong Shading
Light-source type: Point Light
Light-source type: Spot Light
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 26 / 37
Light Components: Specular
simulate light’s interaction with reflective material,
view angle dependent,
highlight of the light’s color, not material color,
I = cos(β)s · Ks, s controls size of the highlight.
Figure: Specular spheres (Phong vs Blinn-Phong)
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 27 / 37
Phong Lighting
Figure: Taken from lighthouse3d.com
amount of reflected light diminishes with increasing angle,
R = −L + 2 · (N • L) · N,
cos(β) = R • Eye,
all vectors must be in same space, normalized.
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 28 / 37
Blinn-Phong Lighting
Figure: Taken from lighthouse3d.com
amount of reflected light diminishes with increasing angle,
H = L + Eye,
cos(β) = H • N,
all vectors must be in same space, normalized.
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 29 / 37
Outline
1 Basics – Repetition
2 Issues with Normals
3 Lighting
Light Components: Ambient
Light-source type: Directional Light
Shading types: Gouraud Shading
Light Components: Diffuse
Shading types: Flat Shading
Light Components: Ambient plus Diffuse
Light Components: Specular
Light Components: Ambient plus Diffuse plus Specular
Shading types: Phong Shading
Light-source type: Point Light
Light-source type: Spot Light
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 30 / 37
Basic Lighting
ambient, diffuse and specular form the baseline lighting,
I = Ka + cos(α) · Kd + cos(β)s · Ks,
light from various sources can be combined (added),
value should not be outside the [0.0, 1.0] range.
Figure: Ambient + Diffuse + Specular pawns
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 31 / 37
Outline
1 Basics – Repetition
2 Issues with Normals
3 Lighting
Light Components: Ambient
Light-source type: Directional Light
Shading types: Gouraud Shading
Light Components: Diffuse
Shading types: Flat Shading
Light Components: Ambient plus Diffuse
Light Components: Specular
Light Components: Ambient plus Diffuse plus Specular
Shading types: Phong Shading
Light-source type: Point Light
Light-source type: Spot Light
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 32 / 37
Phong Shading
per pixel shading,
smooth lighting including details,
interpolation of vertex attributes (normal, eye, light).
Figure: Per pixel lighting pawns
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 33 / 37
Outline
1 Basics – Repetition
2 Issues with Normals
3 Lighting
Light Components: Ambient
Light-source type: Directional Light
Shading types: Gouraud Shading
Light Components: Diffuse
Shading types: Flat Shading
Light Components: Ambient plus Diffuse
Light Components: Specular
Light Components: Ambient plus Diffuse plus Specular
Shading types: Phong Shading
Light-source type: Point Light
Light-source type: Spot Light
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 34 / 37
Light-source type: Point Light
light source inside the scene,
defined by a position vector (all directions),
can represent e.g. a lightbulb.
Figure: Taken from lighthouse3d.com Point light pawns.
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 35 / 37
Outline
1 Basics – Repetition
2 Issues with Normals
3 Lighting
Light Components: Ambient
Light-source type: Directional Light
Shading types: Gouraud Shading
Light Components: Diffuse
Shading types: Flat Shading
Light Components: Ambient plus Diffuse
Light Components: Specular
Light Components: Ambient plus Diffuse plus Specular
Shading types: Phong Shading
Light-source type: Point Light
Light-source type: Spot Light
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 36 / 37
Light-source type: Spot Light
light source inside the scene,
only a directed cone is illuminated,
defined by a position vector, direction vector and angle,
can represent e.g. a flashlight.
Figure: Spot light pawns
PV227 – GPU Rendering (FI MUNI) Lesson 4 – Math / Lighting 6. 10. 2015 37 / 37