2 votes

Find a non-degenerate vector perpendicular to a given 3D vector, general solution

If whatever tentative definition of "nice" should imply that $\vec{r} \mapsto \vec{v} = \vec{v}(\vec{r})$ is continuous then this is impossible to achieve for all $\vec{r} \in \mathbf{R}^3 - ...
ronno's user avatar
  • 11k
1 vote

Let $P$ be a point in the first quadrant that lies on the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$.

Let $P = (x_1,y_1)$ lie on the hyperbola. The normal vector at $P$ is along $ g = [ \dfrac{x_1}{a^2} , -\dfrac{y_1}{b^2} ] $ Therefore the equation of tangent line at $P$ is $ \dfrac{x_1 (x - x_1)}{a^...
of course's user avatar
  • 20.9k
1 vote

Change of Basis in Game Development

Simply: change of basis twice is not quite what you want to do. Let the canonical basis (i.e. that with matrix $1$) be $W$ i.e. $$R\times X_R = B\times X_B = C\times X_C = X_W$$ You can see that all ...
Soundwave's user avatar
  • 423
1 vote

How duality works with polynomials?

Regarding your second question What can be the dual base of $C$? You used the correct definition of dual basis We have $c_i^*(c_j)=\delta_{i,j}$ Hence the dual basis for $C$ would be $(c_1^*,c_2^*,...
ehceb's user avatar
  • 88
1 vote

Is the metric in normal coordinates constant for flat manifolds?

Working locally throughout, we can regard normal coordinates as maps $\varphi:\mathbb{R}^n\to M$ defined by $$ \varphi(x^1,\cdots,x^n)=\exp_p(x^ie_i) $$ where $e_1,\cdots,e_n$ is an orthonormal basis ...
Kajelad's user avatar
  • 14.9k
1 vote

To find the conic touched by chords on contact from a parabola.

You start out fine; the polar of $(t^2,2t)$ is $x t^2+4y2t+2(x+t^2)+3=0.$ Then the trick to get the envelope is to take the partial derivative wrt $t,$ it is $2tx+8y+4t=0.$ Eliminating $t$ from the ...
Jan-Magnus Økland's user avatar
1 vote

Visualisation Tool: Coordinate Systems

Maybe geogebra is what you're looking for or TikZ with LaTeX, on wikipedia all sorts of tools can be used, there are no specific tools for this.
chaise's user avatar
  • 26
1 vote

Intersection of line and lattice points

"In practice" your line does not have zero width. Since a line with irrational slope comes as close as you wish to many lattice points the fuzziness of the pencil lead (or the pixel width on ...
mathemaster's user avatar

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