0
votes
0answers
21 views

How to properly clamp Beckmann Distribution

I am trying to implement the Cook-Torrance Microfacet BRDF shading model and I am having some trouble with the Beckmann Distribution: Beckmann Distribution with width parameter ...
3
votes
6answers
314 views

What is larger: The inside or the outside of the infinite circle? [closed]

Assume a circle with radius $R$ in a plane. Let $R$ go to infinity. What is larger: The inside or the outside of the circle? EDIT My naive way of thinking about "largeness" was just to compare ...
15
votes
3answers
2k views

Is an infinite line the same thing as an infinite circle?

Imagine that you are sitting next to a line that extends infinitely in both directions. Is it possible to distinguish it from an infinite circle? From my poor understanding of topology, I would ...
2
votes
1answer
58 views

Credit Given - Geometricly Modeling Infinity with 3 planes and 9 circles - Ratio of Circles

Refer to the attached diagram sketch to help visualize the equation. I am requesting help with an interesting math problem. Basically, I am diagraming infinity using three planes. These planes ...
0
votes
0answers
50 views

Number of ways to cut a square

How many ways are there to cut the unit square into two pieces? And how many ways are there if the two pieces must have equal area? Some special cases: A. If the cut is required to be a horizontal ...
1
vote
1answer
64 views

n-Ball Volume and surface with $n \rightarrow \infty$

I am thinking about something I just read: The volume of the n-ball is given by $V_n(r) = \frac{\pi^{n/2}}{\Gamma (\frac n 2 + 1)}r^n$ and its surface area is $S_n(r) = \frac{\pi^{n/2}}{\Gamma (\frac ...
4
votes
2answers
3k views

Can parallel lines meet? [duplicate]

Can parallel lines meet? There is a person that takes a calculus course with us, and every time we ask him for something he answers us with I'll do it when two parallel lines meet each other. So I ...
8
votes
5answers
7k views

Prove that the distance between a black and a white dot is one

I just read this article about some tough interview questions. One of the questions (allegedly given in an interview for a Technology Analyst position in Goldman Sachs) was: There are infinite ...
1
vote
0answers
46 views

Meaningful measures for comparing infinite dimensional geometric objects

I have two infinite-dimensional convex polytopes, call them $A$ and $B$. I know that $B$ is completely contained within $A$, and I want to say something meaningful about their relative sizes. From ...
2
votes
3answers
536 views

Number of points on line segment

I know the line segment have a infinite number of points, but i know that exist different kinds of infinity ( $\aleph_0 $). My question is there same number of points on segment of line and entire ...
2
votes
2answers
105 views

Why do geometric sets such as $(\infty, x]$ never have infinity included?

I have a question about the use of infinity and geometric sets. Say I am trying to graph an equation, and the result is all values greater than or equal to, say, $3$. From what I've seen, the proper ...
15
votes
4answers
3k views

Two paradoxes: $\pi = 2$ and $\sqrt 2 = 2$ [duplicate]

Possible Duplicate: Is value of $\pi = 4$? Can anyone explain how to properly resolve two paradoxes in this YouTube video by James Tanton?
6
votes
4answers
3k views

Can a circle truly exist?

Is a circle more impossible than any other geometrical shape? Is a circle is just an infinitely-sided equilateral parallelogram? Wikipedia says... A circle is a simple shape of Euclidean geometry ...
6
votes
3answers
469 views

How is it that this shape can converge to what looks like a triangle but has a different perimeter?

I had this strange notion some time ago, and I recently wrote a blog post about it, as a mere curiosity. I don't really consider it a "serious" mathematical question; but out of interest, I wondered ...
3
votes
1answer
168 views

Proof whether or not 1/k by 1/(k+1) rectangles fit inside a unit square

I am reading Concrete Mathematics and came across an interesting problem, number 37 of chapter 2. The answers to exercises lists no known answer to this problem: Will all the 1/k by 1/(k+1) ...
4
votes
2answers
342 views

Is the number of circles in the Apollonian gasket countable?

Is it correct to say that the number of circles in an Apollonian gasket is countable becuase we can form a correspondence with a Cantor set, as their methods of construction are similar? What about ...