Ray Intersect - Introduction to Computer Graphics - Lecture Slides, Slides of Computer Graphics

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Lecture 17:

Ray Intersect

Review

• In ray casting, there are three steps:

– Step 1: Generating a Ray

• A. Find the ray in world coordinate system
• B. Find the ray in camera coordinate system and transform it

to world coordinate

– Step 2: Intersection

• Intersects the ray with all objects in the world
• Return the nearest point on the nearest object

– Step 3: Coloring the Pixel

• Direct illumination (Assignment 5: intersect)
• Indirect illumination (Assignment 6: recursive ray tracer)

Step 2: Ray-Object Intersection

(Implicit Objects)

• At what points (if any) does the ray intersect an object?
• Points on a ray have form P + t d
  • t is any nonnegative real
• A surface point Q lying on an object has the property that f(Q) = 0
• Combining, we want to know “For which values of t is f(P + t d ) = 0 ?”
• We are solving a system of simultaneous equations in x , y (in 2D) or x , y , z (in 3D)

Step 2: Ray-Object Intersection

(Implicit Objects)

2D ray-circle intersection example

• Consider eye-point P = (-3, 1), direction vector d = (.8, -.6) and the unit circle:

f(x,y) = x^2 + y^2 – R^2

• A typical point of the ray is:

Q = P + t d = (-3,1) + t (.8,-.6) = (-3 + .8t,1 - .6 t )

• Plugging this into the equation of the circle:

f (Q) = f (-3 + .8t,1 - .6 t ) = (-3+.8 t )^2 + (1-.6 t )^2 - 1

• Expanding, we get:

9 – 4.8 t + .64 t^2 + 1 – 1.2 t + .36 t^2 - 1

• Setting this to zero, we get:

t^2 – 6 t + 9 = 0

Step 2: Ray-Object Intersection

2D ray-circle intersection example (continued)^ (Implicit Objects)

• Generalizing:
  • we can take an arbitrary implicit surface with equation f(Q) = 0 , a ray P + t d , and plug the latter into the former:

f ( P + t d ) = 0

• The result, after some algebra, is an equation with t as unknown
• We then solve for t , analytically or numerically

Step 2: Ray-Object Intersection

• For cylindrical objects, the implicit equation^ (Implicit Objects)

𝑓(𝑥, 𝑦, 𝑧) = 𝑥^2 + 𝑧^2 – 1 = 0

in 3-space defines an infinite cylinder of unit radius, running along the y-axis

• Usually, it’s more useful to work with finite objects
  • e.g. a unit cylinder truncated with the limits: Cylinder Body: 𝑥^2 + 𝑧^2 – 1 = 0, −1 ≤ 𝑦 ≤ 1
• But how do we implicitly define cylinder “caps?”
• The cap is the inside of the cylinder at the y extrema of the cylinder (a.k.a. a circle)
  • cylinder caps top: 𝑥^2 + 𝑧^2 – 1 ≤ 0, 𝑦 = 1

bottom: 𝑥^2 + 𝑧^2 – 1 ≤ 0, 𝑦 = −

Cap

Step 2: Ray-Object Intersection

(Implicit Objects)

Implicit surface strategy summary

• Substitute ray ( P + t d ) into implicit surface equations and solve for t
  • smallest non-negative t -value is the closest surface you see from eye point
• For complicated objects (not defined by a single equation), write out a set of equalities and inequalities and then code individual surfaces as cases…
• Latter approach can be generalized cleverly to handle all sorts of complex combinations of objects - constructive Solid Geometry (CSG), where objects are stored as a hierarchy of primitives and 3-D set operations (union, intersection, difference) - “blobby objects”, which are implicit surfaces defined by sums of implicit equations (F(x,y,z)= 0 )

CSG!

Cool Blob!

Step 2: Ray-Object Intersection

• So far, we have seen how intersection works if

the object is in the world coordinate, and its

parameters are simple

– That is, the object sits at (0, 0, 0), and has unit radius

• In real examples, the objects have often been

translated, rotated, and scaled.

• How would we apply the same solution to these

objects?

Step 2: Ray-Object Intersection (Object

Coord)

• Let the world-space intersection point be defined as MQ , where Q is a point in object space: 𝑃 + 𝑡𝒅 = 𝑀𝑄 𝑀−1^ ⋅ 𝑃 + 𝑡𝒅 = 𝑄 𝑀−1𝑃 + 𝑡𝑀−1𝒅 = 𝑄 Let 𝑃 = 𝑀−1𝑃, 𝒅 = 𝑀−1𝒅
• If 𝑓 𝑥, 𝑦, 𝑧 is the equation of the untransformed object, we just have to solve 𝑓 𝑃 + 𝑡𝒅 = 0
• note: 𝒅 is probably not a unit vector
• the parameter t along this vector and its world space counterpart always have the same value.
• normalizing 𝒅 would alter this relationship. Do NOT normalize 𝒅. Why??

(0,0,0)

MQ

P  t d

f ( x , y , z ) 0

d^11 d

~ ~   P  t  M P  tM

Q

f x y z 

Step 2: Ray-Object Intersection (Object

Coord)

• In the example, we already know the matrix M, which transforms the

object from object coordinate into world coordinate.

• Recall, M is the matrix that translate/rotate/scales a unit object into the desired location/orientation/size
• We are just finding the inverse of M, which will bring the ray from world

coordinate back to object coordinate.

• The advantage is obvious, math is easier in the object coordinate space!
• As mentioned before, you need to find the smallest t-value from the

intersection test, you can use t in two ways:

• 𝑃 + 𝑡𝒅 gives the world coordinate location of the intersection point
• 𝑃 + 𝑡𝒅 is the corresponding intersection point in object coordinate
• Note that the t value doesn’t change between the two coordinate spaces!