User:Zachgildersleeve

This is a page created for CS6640 Human Computer Interaction at the University of Utah.

About Me
I am a second year masters student at the U of U's School of Computing. Here I am:

My homepage has more photos and projects (completed and works in progress)

I recently started a blog to organize food recipes of what I eat for dinner. I don’t post every time I eat spaghetti and Bolognese, just the nicer stuff. That’s not to say that spaghetti and Bolognese isn’t nice.

Visit Garlic Is For Heroes

$$\mathbf{\gamma}(t) = \sum_{i=0}^n \mathbf{P}_i\mathbf{\theta}_{i,n}(t),\quad t\in[0,1]$$

where

$$\mathbf{\theta}_{i,n}(t) = {n\choose i} t^i (1-t)^{n-i},\quad i=0,\ldots n$$

and

$${n\choose i} = \frac{n!}{i!(n-i)!}$$

the full expansion takes the form:

$$\mathbf{\gamma}(t) = \mathbf{P}_0{\frac{n!}{0!(n-0)!}} t^0 (1-t)^{n-0} + \mathbf{P}_1{\frac{n!}{1!(n-1)!}} t^1 (1-t)^{n-1} + \ldots + \mathbf{P}_{n-1}{\frac{n!}{n-1!(n-n-1)!}} t^{n-1} (1-t)^{n-n-1} + \mathbf{P}_n{\frac{n!}{n!(n-n)!}} t^n (1-t)^{n-n}$$

$$\mathbf{\gamma}_1(1) = \mathbf{\gamma}_2(0)\implies\mathbf{P}_n = \mathbf{Q}_0$$

$$\mathbf{\gamma}_1^{}(1) = \mathbf{\gamma}_2^{}(0);$$

$$\mathbf{\gamma}_1^{}(1) = \mathbf{C}\mathbf{\gamma}_2^{'}(0) + k\mathbf{\gamma}_2^{}(0)$$

$$\mathbf{P}_n - \mathbf{P}_{n-1} = k(\mathbf{Q}_1 - \mathbf{P}_n)\implies\frac{1}{k}\mathbf{P}_n - \frac{1}{k}\mathbf{P}_{n-1} + \mathbf{P}_n = \mathbf{Q}_1$$

$$n(\mathbf{P}_n - \mathbf{P}_{n-1}) = m(\mathbf{Q}_1 - \mathbf{P}_n)\implies\frac{n}{m}\mathbf{P}_n - \frac{n}{m}\mathbf{P}_{n-1} + \mathbf{P}_n = \mathbf{Q}_1$$

$$(n-1)(n)(\mathbf{P}_n - 2\mathbf{P}_{n-1} + \mathbf{P}_{n-2}) = (m-1)(m)(\mathbf{Q}_2 - 2\mathbf{Q}_1 + \mathbf{Q}_0)$$

$$\frac{(n-1)(n)}{(m-1)(m)}(\mathbf{P}_n - 2\mathbf{P}_{n-1} + \mathbf{P}_{n-2}) - \mathbf{P}_n + 2\mathbf{Q}_1 = \mathbf{Q}_2$$