triangle. Hit beams must continue KD tree traversal until its hit distance is fully contained by one **of** the beam’s tmin planes **and** all **of** its tmax planes. Outline Motivation Beam Tracing Algorithm Beam-Triangle Intersection Beam-KD-tree/ **volumes** [Laine et al. 2005, Lehtinen et al. 2006] Comparison Setup Use our optimized ray tracer for comparison Not clear how to adapt **frustum** methods/ray packets 512x512 resolution (linear scaling with res) Our kd-tree builder (basic **surface** **area** heuristic/

shapes work out the **area** **of** complex shapes made from a combination **of** known shapes work out the **area** **of** segments **of** circles work out **volumes** **of** **frustums** **of** cones work out **volumes** **of** **frustums** **of** pyramids calculate the **surface** **area** **of** compound solids constructed from cubes, cuboids, cones, pyramids, cylinders, spheres **and** hemispheres solve real life problems using known solid shapes Unit 3 – Perimeter, **Area** **and** VolumeUnit 3 – Perimeter, **Area** **and** **Volume** (Slide 3 **of** 3) View previous page Unit/

shapes work out the **area** **of** complex shapes made from a combination **of** known shapes work out the **area** **of** segments **of** circles work out **volumes** **of** **frustums** **of** cones work out **volumes** **of** **frustums** **of** pyramids calculate the **surface** **area** **of** compound solids constructed from cubes, cuboids, cones, pyramids, cylinders, spheres **and** hemispheres solve real life problems using known solid shapes Unit 3 – Perimeter, **Area** **and** VolumeUnit 3 – Perimeter, **Area** **and** **Volume** (Slide 3 **of** 3) View previous page Unit/

**volume**, curvature, **surface** normal, etc... 46 SSD99 Tutorial Triangle quality measures o A TIN should have triangles as much compact as possible o Elongated triangles (slivers) cause numerical errors **and** unpleasant visual effects o Compactness **of** a triangle [Gueziec, 1995]: where A is the **area** **and**/with: maccuracy decreasing linearly with distance from the viewpoint mregion **of** interest coincident with the view **frustum** 159 SSD99 Tutorial...VARIANT......Multiresolution visualization... Uniform resolution: 12701 /

channels… Use XNA RenderTarget2D Writeable **surface**, can convert to texture Easy to use Creating the Shadow Map Matrices Need light’s view, projection matrices Choice in projection Ortho best for large, directional lights (i.e. sun) LightProj = Matrix.CreateOrthographic(w, h, n, f) w, h are width **and** height **of** view **volume** n, f are near **and** far planes **of** the view **volume** Recommend calculating these instead/

al., 00: Frame Coherent **Volume** Rendering. New Algorithm: Overview Main Idea: - Compute a single intersection **of** the ray with an object, if possible - If not possible -> ordinary ray shooting algorithm First frame: use ordinary ray shooting, remember intersection points in 3D space Next frames: reproject points, check if we can decide on intersection objects Properties: intersection points **and** **surface** normals compute correctly (unlike in/

**surface** in scene To save time, draw only **surfaces** we see **Surfaces** we cannot see **and** their elimination methods: Occluded **surfaces**: hidden **surface** removal (visibility) Back faces: back face culling Faces outside view **volume**/each polygon { for each pixel (x,y) inside the polygon projection **area** { if (z_polygon_pixel(x,y) > depth_buffer(x,y) ) { /**Frustum** Culling Remove objects that are outside the viewing **frustum** Done by 3D clipping algorithm (e.g. Liang-Barsky) Ray Tracing Ray tracing is another example **of**/

Shadow **Volumes** MIT EECS 6.837, Durand **and** Cutler Cast Shadows on Planar **Surfaces** Draw the object primitives a second time, projected to the ground plane MIT EECS 6.837, Durand **and** Cutler Limitations **of** /**volumes** 2. Clip the shadow **volumes** to the view **frustum** 3. "Z-Fail" shadow **volumes** 0 MIT EECS 6.837, Durand **and** Cutler 1. Test Eye with Respect to **Volumes** Adjust initial counter value Expensive 0 +1 0 MIT EECS 6.837, Durand **and** Cutler 2. Clip the Shadow **Volumes** Clip the shadow **volumes** to the view **frustum** **and**/

frame – eye at origin Allows use **of** canonical viewing procedures Type **of** projection (parallel or perspective) **and** part **of** world to image (clipping or view **volume**) Normalization lets clip against simple cube regardless **of** projection Delay final projection until end –Important for hidden-**surface** removal to retain depth information as long as possible OpenGL Clip Space In OpenGL clip space, viewing **frustum** is mapped to a cube that/

**surfaces** **and** **volume** data 3 LODMODELSLODMODELS VIS98 Tutorial LOD Models o Layered models mdescription **of** a sequence **of** few meshes each **of** which represents an object at a different resolution o Multiresolution models mdescription **of** a virtually continuous set **of**/set describes the region **of** interest **of** the query Examples **of** focus sets: mPoint : point location query mLine/polyline : segment/line interference query mRegion : window query, region interference query mVolume: view **frustum** VIS98 Tutorial...Spatial /

the **volume** **of** the cone The theorems **of** Pappus **and** Guldinus are used to find the **surfaces** **area** **and** **volume** **of** any object **of** revolution provided the generating curves **and** **areas** do not cross the axis they are rotated 9.4 Theorems **of** Pappus **and** Guldinus **Surface** **Area** **Area** **of** a **surface** **of** revolution = product **of** length **of** the curve **and** distance traveled by the centroid in generating the **surface** **area** 9.4 Theorems **of** Pappus **and** Guldinus **Volume** **Volume** **of** a body **of** revolution = product **of** generating **area** **and** distance/

) Alternative to glFrustum(..). Creates a matrix for an perspective viewing **frustum** **and** multiplies the current matrix by it. Alters the Projection matrix. Note: fovy is the field **of** view (fov) angle between the top **and** bottom planes **of** the clipping **volume**. aspect is the aspect ratio Perspective Projection Commands Enabling GL Features Features – lighting, hidden-**surface** removal, texture mapping, etc… Each feature will slow down the rendering/

high fineness means more strength,more workability,**and** more interaction The exceeded fineness cause – Increases in **surface** **area** **and** more released reaction heat – More cost **of** KLinker – Increase the cement **volume** decrees. Fineness test. This test is done to verify the standard **of** grinding **of** cement. We know that rate **of** hydration **and** hydrolysis **of** cement, depend, upon its fineness **and** thus testing **of** the fineness **of** the cement is an essential feature. Fineness/

complex to analyze easily **Area** Subdivision Four possible relationships between polygon **surfaces** **and** a rectangular section **of** the viewing plane Terminating criteria –Case 1: An **area** has no inside, overlapping, or surrounding **surfaces** (all **surfaces** are ourside the **area**) –Case 2: An **area** has only one inside, overlapping or surrounding **surfaces** –Case 3: An **area** has one surrounding **surface** that obscures all other **surfaces** within the **area** boundaries Octrees Visible-**surface** identification is accomplished by/

the slower the heat transfer rate. A cone **frustum** is conducting heat at steady state with no generation A cone **frustum** is conducting heat at steady state with no generation. If the **frustum** is insulated as shown, the heat flux in/as **volume**/**surface** **area**, **and** k is thermal conductivity. To keep the Biot number small would require either a small convective heat transfer coefficient or characteristic length or large thermal conductivity. -Effects **of** density on heat transfer would depend on the material **and**, /

**of** the cone is h The curved **surface** **area** **of** the cone **and** the curved **surface** **area** **of** the hemisphere are equal (a) Show that l = 2r (2) (b) Find the perpendicular height, h, **of** the cone in terms **of** r. (2) (c) Find the ratio **of** the **volumes** **of** the cone **and** the hemisphere. (2) Be prepared to use algebra to solve problems involving **volume** **and** **surface** **area** If the curved **surfaces** **of**/**volume** **of** the whole cone including the missing top which shouldn’t really be there is 1/3 × π × 62 × 3 = 36π The **volume** **of** the **frustum** /

a **Volume** **of** a **Frustum** A1 h A2 A1 **and** A2 are parallel Perpendicular height between A1 **and** A2 h A2 A1 **and** A2 are parallel **Volume** **of** a Wedge h = Vertical Height a b d h If d=a **Volume** **of** a Prismoid A Prismoid is a solid having for its ends any two parallel plane figures, **and** having plane sides. l A2 A1 Trapezoidal Rule for **Volumes** Prismoidal Formula End **Area**/

Description **of** all object, **surface** **and** light source geometry **and** transformations Lighting model: Computational description **of** object **and** light properties, interaction (reflection, scattering etc.) Synthetic viewpoint (or Camera location): Eye position **and** viewing **frustum** Raster /simulation: Fluid flow visualization for graphics **and** scientific computing One **of** the most well-studied **and** successful uses **of** the GPU. Image processing **and** analysis A very hot **area** **of** research in the GPU world Numerous /

different rated conditions –Calculate engine **volume**, piston speed, **and** cylinder **surface** **area** as a function **of** crank angle –Draw a cylinder schematic **and** identify the bore, stroke, crank radius, TDC, BDC, **and** crank angle List **of** Laboratory Experiments Study **of** Anatomy **of** A Single Cylinder Diesel Engine. Study **of** Anatomy **of** A Multi-cylinder Diesel Engine. Disassembly **and** assembly **of** A single cylinder Diesel Engine. Measurement **of** Valve Timing **of** A Single Cylinder Diesel Engine. Performance/

boundaries **of** a closed **surface**. By choosing an arbitrary interior point, the complete interior **of** the **surface** will be filled with the color **of** the user’s choice. Interactive Filling, cont’d Definition: An **area** or/**volume** so that the centerline **of** the **frustum** is perpendicular to the view plane 2.Scale the view **volume** with a scaling factor that depends on 1/z. A shear operation is to align a general perspective view **volume** with the projection window. The transformation involves a combination **of** z-axis shear **and**/

**of** California, San Diego Evolution **of** **volume** graphics **Volume** graphics today… …is where **surface** graphics was 15 years ago –We are at the start **of** a transition from technology to tool What enabled story telling **and** games for **surface** graphics? What might do the same for **volume**/ a small **volume**… History Rendering Authoring Data source: Product literature SAN DIEGO SUPERCOMPUTER CENTER University **of** California, San Diego What resolution can we see? Eye’s lens focuses light onto retina –Fovea = focus **area** = center /

viewing procedures Type **of** projection (parallel or perspective) **and** part **of** world to image (clipping or view **volume**) Normalization lets clip against simple cube regardless **of** projection Delay final projection until end –Important for hidden-**surface** removal to retain depth information as long as possible OpenGL Clip Space Overview, model-view-projection In OpenGL clip space, viewing **frustum** is mapped to a cube that extends from -1/

the **volume** **of** the cone The theorems **of** Pappus **and** Guldinus are used to find the **surfaces** **area** **and** **volume** **of** any object **of** revolution provided the generating curves **and** **areas** do not cross the axis they are rotated 9.4 Theorems **of** Pappus **and** Guldinus **Surface** **Area** **Area** **of** a **surface** **of** revolution = product **of** length **of** the curve **and** distance traveled by the centroid in generating the **surface** **area** 9.4 Theorems **of** Pappus **and** Guldinus **Volume** **Volume** **of** a body **of** revolution = product **of** generating **area** **and** distance/

outer **surface** **of** the **volume** (T p ) Sampling Constant interval (easy but may lose info) Curvature Based (harder, more accurate) Intersect T p with T s to find collisions Type 2 Collisions Compute NCP at some sampling density Similar sampling issues as before Triangulate each NCP **and** intersect with T s Add these triangles to T p Path correction In general, find problem **areas** **and** push/

using hardware texture mapping 3/16/04James R. McGirr3 Motivation Many **areas** in medicine, computational physics, biology, etc, deal with large volumetric data sets that demand adequate visualization. Direct **Volume** Rendering – Each point in space is assigned a density for the emission **and** absorption **of** light **and** the **volume** renderer computes the light reaching the eye along viewing rays. Can be done efficiently using texture mapping/

sample database downloaded from NIMA contained 544 MB **of** raw data. The process **of** removing extraneous objects from being drawn is known as culling. There are many methods **of** culling commonly used in 3D graphics, including: View-**Frustum** Culling Bounding-**Volume** Culling Bounding-**Volume** Hierarchies Backface Culling Cells **and** Portals Levels **of** Detail VPF Topology Levels Method **of** changing the level **of** detail in the VPF data. Included in the/

complex to analyze easily **Area** Subdivision Four possible relationships between polygon **surfaces** **and** a rectangular section **of** the viewing plane Terminating criteria –Case 1: An **area** has no inside, overlapping, or surrounding **surfaces** (all **surfaces** are ourside the **area**) –Case 2: An **area** has only one inside, overlapping or surrounding **surfaces** –Case 3: An **area** has one surrounding **surface** that obscures all other **surfaces** within the **area** boundaries Octrees Visible-**surface** identification is accomplished by/

= 17.5 – 3.5 = 14cm **Volume** **of** wood = **volume** **of** hemisphere + **volume** **of** cone 2 2 A cup is in the form **of** a hemisphere surmounted by a cylinder A cup is in the form **of** a hemisphere surmounted by a cylinder. The height **of** the cylindrical portion is 8 cm **and** the total height **of** the cup is 11.5 cm. Find the total **surface** **area** **of** the cup. Cylindrical part Height = 8/

to an identity matrix 23 Enabling GL Features Features – lighting, hidden-**surface** removal, texture mapping, etc… Each feature will slow down the rendering /**frustum** **and** multiplies the current matrix by it. Alters the Projection matrix. 42 Note: fovy is the field **of** view (fov) angle between the top **and** bottom planes **of** the clipping **volume**/ Defines an **area** **of** the window into which the final image is mapped (x, y) specifies the lower-left corner **of** the viewport (width, height) specifies the size **of** the viewport /

all the way through. Its **volume** is found by working out the **area** **of** the end face, **and** multiplying this by the length **of** the shape. Ex 30.5 Q1,2 **and** 3 **Surface** **area** **of** a prism Total **surface** **area** **of** a prism is found by adding up the **area** **of** each face. For example, a triangular prism is made up **of** 2 identical triangular faces **and** 3 rectangular ones. **Surface** **area** **of** a cylinder h Ex30.6/

planes to define max **and** min distance to viewer Introduction to 3D clipping 265 In 3D a clip **volume** needs to be defined Example for “ perspective ” 3D Introduction to 3D clipping 266 Result is a finite clip **volume** **of** space This shape is a named a **frustum** Introduction to 3D clipping 267 3-D Extension **of** 2-D Cohen-Sutherland Algorithm, Outcode **of** six bits. A bit/

= 20 ft 51032 ft2 3533 ft2 1649 ft2 **Frustum** **of** a Right Circular Cone The part **of** a right circular cone between the base **and** a plane parallel to the base whose distance from the base is less than the height **of** the cone. Height: h Radius **of** bases: r, R Slant height: s Lateral **surface** **area**: S Total **surface** **area**: T **Volume**: V s = sqrt([R-r]2+h2) S/

**area** light ??[Moeller02] [Blinn76] [Heidrich00] [Crow77] [Ashikhmin02] our technique Non-Interactive[Matusik02] Motivation Better light integration **and** transport dynamic, **area** lights self-shadowing interreflections For diffuse **and** glossy **surfaces** At real-time rates point light **area** light **area** lighting, no shadows **area**/ the eye’s point-**of**-view For each rasterized fragment determine fragment’s XYZ position relative to the light this light position should be setup to match the **frustum** used to create the depth/

. Color is calculated for each vertex **and** interpolated across the polygon Phong Shading The normal vectors are interpolated across the **surface** **of** the polygon The color **of** each point within the polygon is calculated from its corresponding normal vector Polygon shading techniques compared Viewing **frustum** Segment **of** the 3D world to be rendered Objects outside the viewing **volume** are ignored. Hidden **surface** determination Not all objects inside the/

all the possible light paths to the **surface** – the more paths, the greater the level **of** realism Radiosity **and** Ray Tracing Thus global illumination is divided into two main approaches Radiosity – calculation **of** light as radiation transfer through **volume** **of** air Ray Tracing – calculation **of** light as a set **of** discrete ray samples through a **volume** We have covered the general idea **of** ray tracing Now we illustrate the main/

the **volume** **of** a pyramidal **frustum**. The first documented systematic technique capable **of** determining integrals is the method **of** exhaustion **of** Eudoxus (circa 370 BC), which sought to find **areas** **and** **volumes** by breaking them up into an infinite number **of** shapes for which the **area** or **volume** was known. This method was further developed **and** employed by Archimedes (287 BC - 212 BC) **and** used to calculate **areas** for parabolas **and** an approximation to the **area** **of** a/

**surface** **area**. Minimizing this **area** increases the efficiency **of** any intersection algorithm as a rejection is never slower to compute than an intersection. It is often better to minimize the **volume** **of** BV for collision detection algorithms. 14 AABB **and** k-DOP Creation AABB The simplest bounding **volumes** to create is an AABB. Take the minimum **and** maximum extents **of** the set **of**/it misses. A generalized **of** the slabs method can be used to compute the intersection **of** a ray with a k-DOP, **frustum**, or any convex /

Example view **volume** left = -1/**Of** Color elements **of** color: 49 Basics **of** Color physics illumination electromagnetic spectra reflection material properties **surface** geometry **and** microgeometry polished versus matte versus brushed perception physiology **and** neurophysiology perceptual psychology 50 Light Sources common light sources differ in kind **of**/**of** time, per unit solid angle, **and** per unit projected **area** **of** the source (related to the luminance **of** the source) lightness/brightness: perceived intensity **of**/

6 Review: Projection Normalization warp perspective view **volume** to orthogonal view **volume** render all scenes with orthographic projection! aka /the field **of** view / **frustum** solved by clipping – polygon is backfacing solved by backface culling – polygon is occluded by object(s) nearer the viewpoint solved by hidden **surface** removal / – all positive, unit **area** – Y is luminance Backstory: Spectral Sensitivity Curve 80 Review: CIE Chromaticity Diagram **and** Gamuts plane **of** equal brightness showing chromaticity gamut/

until: –rectangle contains part **of** 1 projected **surface** –rectangle contains part **of** no **surface** –rectangle is size **of** pixel 6/14/2015©Zachary Wartell **Area**-Subdivision Method We need tests that can quickly determine tell if current **area** is part **of** one **surface** or if further subdivision is needed. Four cases for relation between **surface** **and** rectangular **area**: surrounding **surface** overlapping **surface** inside **surface** outside **surface** 6/14/2015©Zachary Wartell **Area**-Subdivision: Stopping Conditions Recursive subdivision/

**of** Aliasing Wilf LaLonde ©2012 Comp 4501 An Example **of** Acne Wilf LaLonde ©2012 Comp 4501 Reduce **areas** **of** extreme mag- **and** mini- fications (increase resolution or change the engine architecture) Perspective shadow maps => warp the shadow space to the view space Cascaded shadow maps => use different shadow maps in different parts **of** the **frustum**/Pixel is blocked by something... } Wilf LaLonde ©2012 Comp 4501 This is Called **Surface** Acne or Z-Fighting Wilf LaLonde ©2012 Comp 4501 Eliminating Acne From “Real-time /

& 12.5 **Volume** **of** Prisms & Cylinders olume **of** Pyramids & Cones Go over Quizzes **Volume** **of** Prisms **and** Cylinders What physical process did we relate to **surface** **area**? What about **volume**? Formula for the **Volume** **of** a Prism **and** Cylinder Prism Cylinder REMEMBER: B – represents the **area** **of** one base h – represents the height **of** the prism or cylinder cross section Example 1: A triangular prism has a height **of** 10 cm. The lengths **of** the base edges/

– Leaves represent convex polytope Some solid, some empty Some have infinite **area**/**volume** Works best for indoor environments (flat, man-made **surfaces**) Used by original Doom **and** Quake – Still used today (Source, Unreal, Call **of** Duty IW engine) a b c d e f g h b / 3 – Seen from portal 2 Don’t draw room 4 – Not seen from portal 3 Player View **frustum** 1 2 3 4 Portal pros **and** cons Pros – Easy collision detection **and** raytracing – Can be used for dynamic objects (unlike BSP trees) – Portals can point to non-adjacent/

to their projection along the selected axis, resulting in a balanced tree. Minimize the sum **of** the **volumes** (or **surface** **areas**) **of** the child **volumes** Minimize the maximum **volume** (**surface** **area**) **of** the child **volumes** Minimize the **volume** (**surface** **area**) **of** the intersection **of** the child **volumes** Maximize the separation **of** child **volumes** Divide primitives equally between the child **volumes** Combinations **of** the previous strategies. 15 Top-down Construction Stopping criteria The node contains just a single/

**of** view / **frustum** solved by clipping polygon is backfacing solved by backface culling polygon is occluded by object(s) nearer the viewpoint solved by hidden **surface** removal for efficiency reasons, we want to avoid spending work on polygons outside field **of** view or backfacing for efficiency **and**/projection transform each vertex maintains z coordinate relative to eye point can do this with canonical viewing **volumes** 108 The Z-Buffer Algorithm augment color framebuffer with Z-buffer or depth buffer which stores Z/

**of** view / **frustum** clipping –polygon is backfacing backface culling –polygon is occluded by object(s) nearer the viewpoint hidden **surface** removal Week 10, Mon 3 Nov 03 © Tamara Munzner8 Back-Face Culling on the **surface** **of**/ canonical viewing **volumes** –each vertex/ Z along edges **and** across spans Week/**area**-averaged accumulation buffer z-buffer: one visible **surface** per pixel A-buffer: linked list **of** **surfaces** data for each **surface** includesdata for each **surface** includes RGB, Z, **area**-coverage percentage,...RGB, Z, **area**/

. It is the amount **of** flux from a point source contained in a small angular **volume**. Steradian (sr): The solid angle subtended at the center **of** a sphere by an **area** on its **surface** numerically equal to the square **of** the radius When the solid/ **of** energy that leaves **surface** i **and** lands on **surface** j Radiosity equation: Radiosity B is the energy per unit **area** leaving the patch **surface** per discrete time interval **and** is the combination **of** emitted **and** reflected energy: B(x) i dA i - total energy leaving a small **area**/

get performance increase as well as quality enhancement The UNIVERSITY **of** NORTH CAROLINA at CHAPEL HILL 6 My Approach **Area** Light Source Occluder **Surface** Hardware Camera View **Frustum** Camera’s View Average Pixels to get Occlusion percentage The UNIVERSITY **of** NORTH CAROLINA at CHAPEL HILL 7 My Approach **Area** Light Source Occluder **Surface** Camera’s View The UNIVERSITY **of** NORTH CAROLINA at CHAPEL HILL 8 My Application The/

Mainly from material in the second half **of** the quarter – will not include material from last part **of** last lecture (**volume** rendering, image-based rendering) Review session/**area** ALSO gets smaller as a cosine fall off! – F att x I x K d x (N L) N V I length = cos(t) Radiance intensity: intensity/solid angle N V Lighting BRDF = Bidirectional Reflectance Distribution Function – Description **of** how the **surface** interacts with incident light **and** emits reflected light – Isotropic Independent **of** absolute incident **and**/

Demonstrative vs Plausible Reasoning Patterns **of** Plausible Inference **Volume** II By G. Polya Princeton Univ. Press 1954 Conjecture Any integer **of** the form 8N+3, where N=1,2,3,… is the sum **of** a square **and** the double **of** a prime N=1 then 8n+3=11 N=2 then 8n+3=19 / example The **area** **of** the lateral **surface** **of** the **frustum** is: Theorem: (R+r) sq rt [(R-r) 2 + h 2 ] Can you check this result by applying to some case you already know? When R=r you get cylinder Consequence B 1 : **Area** is (2 R) h When r=0, **and** h=0 /

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