Tagged Questions
1
vote
1answer
89 views
Projecting a point on a plane through a matrix
I need to render some shadows in opengl, one way to do this is to render your object twice, a first time multiplying it by a special "shadow matrix" that flat your object on a plane generating the ...
-1
votes
1answer
80 views
How to change XYZ axes system into another 'xy' system
I have $3D$ point set lying on a vertical plane. This plane is not parallel to either $X$ or $Y$ axis but makes an angle (say, $\theta$) to $X$ axis. And also it has some ($+$ or $-$) intercept to the ...
0
votes
1answer
48 views
Recovering a conic from a pole-polar pair
Consider a conic section $C$ in $\mathbb{R}^2$. Every point $P$ in the plane has a "dual" (pole-polar duality) line $L$ with respect to $C$ such that lines $PA$ and $PB$ are tangent to $C$, where $L ...
0
votes
0answers
93 views
Full rank decomposition and column space
I have a matrix $B \in \mathbb{R}^{n \times m}$ with $\operatorname{rank}(B)=k$ and full rank decomposition $B = FG'$ where $G$ is an orthonormal basis of the span of $B$ with $G'G=I_k.$ Let $P_{X}$ ...
1
vote
1answer
102 views
Intuitive interpretation of the 3D to 2D mapping
Suppose a 3D configuration of points is given, $X\in\mathbb{R}^{n\times 3}$, and a non-zero matrix $Q\in\mathbb{3\times 2}$, with orthonormal columns. Now, suppose a mapping to 2D is obtained as
...
0
votes
1answer
99 views
Chain Rule and Homogenous Coordinates
I have a vector $\tilde{p} = (x,y,z)$ (homogenous coordinates). The corresponding non-homogenous vector is $p = (x/z, y/z)$.
Now the $\tilde{p}$ is a result of some linear transform $R(\theta)$ of ...
9
votes
2answers
524 views
Augmented Reality Transformation Matrix Optimization
i am a software developer, i'm working on an Augmented Reality system.
I'd like to receive some advice in order to optimize my math model. My program has to be slim and fast.
Here's the situation:
...
