Articles of analytic geometry

4 points in 3-d space (one known and three unknown)

Problem in 3-d space important for computer vision. We have four points: $P_0$ where we know coordinates $(0,0,0)$ and $P_1, P_2, P_3$ where coordinates are unknown. However we know distances between $P_1, P_2, P_3$ (let’s name them $d_{12}, d_{23}, d_{13}$) and unit vectors $v_1, v_2, v_3$, corresponding to the vectors $\overrightarrow{{P_0}{P_1}}, \overrightarrow{{P_0}{P_2}}, \overrightarrow{{P_0}{P_3}}$. How to […]

Conditions for intersection of parabolas?

What are the conditions for the existence of real solutions for the following equations: $$\begin{align} x^2&=a\cdot y+b\\ y^2&=c\cdot x+d\end{align}$$ where $a,b,c,d $ are real numbers. These represent two parabolas; how might we find out the conditions for the existence of $0,2,4$ real solutions of the equations?

Conics definition from Lens formula

Starting with Lens formula directly $$ \frac1u + \frac1u = \frac1f $$ or in its Gauss form: $$ (u-f)(v-f) = f^2, $$ how to recast this into the conics form using definition of eccentricity $$ \frac{PF}{PD} = e\,, $$ at least as an approximation, using geometric optics where $PF,PD$ are focal and directrix distances of […]

Conic Sections and Complex numbers

If $\omega$ is a complex number such that |$\omega$| does not equal 1, then the complex number $$z = \omega + \frac{1}{\omega}$$ describes a conic. The distance between the foci of the conic described by $z$ is? I attempted to rewrite this in cartesian form but to no avail. How do i proceed?

Pair of straight lines

Question: Find the equation of the bisector of the obtuse angle between the lines $x – 2y + 4 = 0$ and $4x – 3y + 2 = 0$. I don’t even know how to proceed here. I know how to find the angle between two lines, but not sure whether that would help in […]

How to transform (rotate) this hyperbola?

Given this hyperbola $x_1^2-x_2^2=1$, how do I transform it into $y_1y_2=1$? When I factor the first equation I get $(x_1+x_2)(x_1-x_2)=1$, so I thought $y_1=(x_1+x_2)$ and $y_2=(x_1-x_2)$. Therefore the matrix must be $ \begin{pmatrix} 1&1\\1&-1 \end{pmatrix} $. But in maple it is just drawn as a straight line. However, it should be a rotated hyperbola. Is […]

Recomputing arc center

Please excuse my poorly drawn doodle here, I’m almost inept at drawing. I’m attempting to compute i2, j2, x2, y2. Knowns: x1, y1, xk, yk, i1, j1, the arc is circular Constraints: resulting arc is circular cartesian co-ordinate system intermediate calculations can be in any co-ord system that makes sense problem may be arbitrarily rotated, […]

$ax+by+cz=d$ is the equation of a plane in space. Show that $d'$ is the distance from the plane to the origin.

This is a 3 part practice question I would like to get some feedback on. I think I have solved the 1st two parts, but I need a little direction for part (c) (the title is Part (a) ) which is repeated here with more detail, a) $ax+by+cz=d$ is the equation of a plane in […]

Rational Point in circle

How many rational point(s) (a point (a, b) is called rational, if a and b are rational numbers) can exist on the circumference of a circle having centre (pie, e)

Is it true that all of the euclidean geometry problem in the IMO(international mathematical olympiad) could all be solve by the analytical geometry?

Is it true that all of the euclidean geometry problem in the IMO(international mathematical olympiad) or even generalize to say that all the plane geometry problem and 3d-geometry could be solve by the analytical geometry/algebraic geometry?