Questions on the use of algebraic techniques to prove geometric theorems.

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5
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1answer
1k views

Formula for curve parallel to a parabola

I have a simple parabola in the form $y = a + bx^2$. I would like to find the formula for a curve which is parallel to this curve by distance $c$. By parallel I mean that there is an equal distance ...
3
votes
3answers
1k views

Canonical to Parametric, Ellipse Equation

I've done some algebra tricks in this derivation and I'm not sure if it's okay to do those things. $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = \cos^2\theta + ...
3
votes
3answers
197 views

How to calculate the x/y coordinate of F in this diagram (geometry)

In the diagram, I've provided, how do I calculate the $x$, $y$ coordinates of $F$ if the points $A$, $B$, $C$ are arbitrary points on a grid? I'm looking for a formula to solve $F's$ $X$ axis and ...
3
votes
1answer
157 views

Simulation of bouncing circles

I want to simulate two circles bouncing off one another. For this I am not sure what I need to calculate. I couldn't find any useful information on the internet, so I have thought long and hard about ...
6
votes
6answers
2k views

Where can I find Linear Algebra in Nature?

I'm a Computer Science major and I've been studying Analytic Geometry and Linear Algebra this semester. Today my teacher gave a hell of an explanation talking about linear systems, quadratic ...
2
votes
1answer
196 views

Easier way to calculate this point besides line intersection?

Given are all points except E, plus |AF| = |DC|. Considering that the lines AB and FE, as well as BC and ED are parallel, is there an easier way to calculate E? Maybe some relation with B? I'd ...
5
votes
1answer
3k views

Analogue of spherical coordinates in $n$-dimensions

What's the analogue to spherical coordinates in $n$-dimensions? For example, for $n=2$ the analogue are polar coordinates $r,\theta$, which are related to the Cartesian coordinates $x_1,x_2$ by ...
2
votes
1answer
285 views

Find increment amount to get from $(x_1, y_1)$ to $(x_2,y_2)$ one dot at a time

If I have two points in positive cartesian coordinates, how do I find: The slope of a line between those points The increment amount to get from $(x_1, y_1)$ to $(x_2, y_2)$ one dot at a time. ...
1
vote
1answer
278 views

Are these sufficient conditions to define an elliptical cone?

I was successful in deriving the equation for an elliptical double-napped cone in rectangular coordinates. All I did was define a line with slope $a$ on the xy-plane, and another line of slope $b$ on ...
1
vote
3answers
535 views

Distance between skew lines - correct method ?

If we have two skew lines in $\mathbb R^3$, $\vec r_{1} = \vec a + \lambda\vec d_1$ and $\vec r_{2} = \vec b + \mu\vec d_2$ then at their closest point, the difference vector $\vec r_2 - \vec r_1$ is ...
1
vote
1answer
285 views

questions from vectors applications

suppose that An boat captain wants to travel due south at 40 knots. If the current is moving northwest at 16 knots, in what direction and magnitude should he work the engine? here is given picture ...
0
votes
1answer
633 views

rotational of polynomials

first of all it is well known that if we rotate (x,y) coordinate by some angle (let's say by A) then new image(x',y') will be related to (x,y) by the following formula ...
5
votes
3answers
2k views

How is the angle between 2 vectors in more than 3 dimensions defined?

I would like to know how the angle between two n-vectors is defined. I mean whether it is unique and how we may compute it (is the inner product a valid method in the n-dimensional space?). I have ...
1
vote
2answers
3k views

rotation by 180 angle

In general I know that if we rotate $(x, y)$ about origin through $180^\circ$ we will get new image $(-x, -y)$, but suppose that we make rotation not about origin but some other point $(a, b)$ does ...
2
votes
1answer
113 views

Finding the location of the end of an arc, knowing the beginning, the arc's length and the radius

I apologise in advance if this is really basic. I have a circle of radius $15$, from which I work out an arc, given an angle of arbitrary value (it's for a computer program). Given that I know the ...
2
votes
2answers
1k views

Finding the intersection of two points and an arbitrary axis

Given two points I would like to find where the line joining them intersects an arbitrary axis. For example, if I had one point $(5, 10)$ and another at $(50, 100)$ I can be sure that somewhere a ...
2
votes
2answers
5k views

How do we prove the rotation matrix in two dimensions not by casework?

I was trying to prove: To carry out a rotation using matrices the point $(x, y)$ to be rotated from the angle, $θ$, where $(x′, y′)$ are the co-ordinates of the point after rotation, and the formulae ...
4
votes
3answers
200 views

What is the most direct way to derive an equation for a parabola from its x and y intercepts?

I have a pair of points at my disposal. One of these points represents the parabola's maximum y-value, which always occurs at x=0. I also have a point which represents the parabola's x-intercept(s). ...
6
votes
5answers
13k views

Determining if an arbitrary point lies inside a triangle defined by three points?

Is there an algorithm available to determine if a point P lies inside a triangle ABC defined as three points A, B, and C? (The three line segments of the triangle can be determined as well as the ...
5
votes
2answers
260 views

Locus and concurrent lines

This will be my first question :-) Let $\mathcal{D}_1$ and $\mathcal{D}_2$ two concurrent lines, and $F$ a point in the plane, and $H$ and $G$ its images by the symmetries of axis $\mathcal{D}_1$ and ...
3
votes
1answer
246 views

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 ...
1
vote
2answers
200 views

Distribution or bounds for maximum Cartesian coordinate sampled from the sufarce of an n-sphere

It's been said that for high dimensions a hypersphere is "nearly all equator". The amount of space near the poles is just ridiculously small. This of course means that from a uniformly random sample ...
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vote
2answers
1k views

Analytic Geometry: Point coordinates, same distance from two points

Given are two points, $P1(x_1, y_2)$ and $P2(x_2, y_2)$, and distance $a$. Now I want to find the two points $T1$ and $T2$. $$d(P1,T1) = d(P2, T1) = a = d(P1,T2) = d(P2, T2)$$ Eg: (T1 and T2 are my ...
1
vote
2answers
184 views

Finding a random vector exactly yay far from another point in 3D space

So I am trying to find a vector a certain distance away from another point ( the distance varies based on an input ) and I've figured out that ...
0
votes
3answers
1k views

find minimum positive angle between two line

i have one question suppose there is given two line by the tangent form $y=3x-2$ and second line $y=5x+3$ we are asked to find smallest positive angle bewteen them in genaral i want to know ...
2
votes
3answers
966 views

Simple analytic geometry - calculate the point coordinates

I have the coordinates from A and B and the distance d. How can I calculate the point C? If the triangle wasn't tilted, it'd be peace of cake. Actually, why is it easier if the triangle isn't ...
7
votes
6answers
3k views

Product of slopes is -1 iff perpendicular proof from first principles

Once again I'm working through Stillwell's Four Pillars of Geometry. I'm on Chapter 3 where he first introduces coordinates. The question reads, 3.5.1 Show that lines of slopes $t_1$ and $t_2$ ...
2
votes
1answer
189 views

How to translated a scaled figure so that a marked point remains fixed?

I've got a bit of a mathematical problem. Basically I have a square, and the width and height are multiplied by $1.02$. Now in the square is a point, which we'll refer to as $(c, d)$, and the origin ...
1
vote
1answer
149 views

Find third point in mapping system

I have two points defined: $A$ and $B$ For both I know $x,y$, longitude, and latitude (gps coordinates). How do I calculate $x,y$ of a third point $C$ when I know its longitude and latitude? I know ...
2
votes
1answer
378 views

Finding the radius of the largest sphere possible between a corner and another sphere

In a 3 dimensional Cartesian plane there is a sphere A that is in the first octant and is tangent to all coordinate planes. Now, imagine we want to find the another sphere B also tangent to all ...
1
vote
1answer
486 views

Trying to pick a random point on sphere end up picking from a lune

I was inspired by this question to play around a little bit (its a weekend). I was pretty confident of my derivation and thought it might be nice to supplement it with a pretty picture. However, ...
10
votes
3answers
2k views

The vertices of an equilateral triangle are shrinking towards each other

For an equilateral triangle ABC of side $a$ vertex A is always moving in the direction of vertex B, which is always moving the direction of vertex C, which is always moving in the direction of vertex ...
6
votes
2answers
3k views

Finding shortest distance from point to plane

I need you guys to check my homework question out if I'm wrong or not... Given point $(1,4,1)$ in need to find the shortest distance between this and the plane $2x_1 - x_2 + x_3 = 5$. So firstly, I ...
5
votes
4answers
1k views

Find the centre of a circle passing through a known point and tangential to two known lines

I am trying to find the centre and radius of a circle passing through a known point, and that is also tangential to two known lines. The only knowns are: Line 1 (x1,y1) (x2,y2) Line 2 (x3,y3) ...
1
vote
3answers
450 views

Questions about Lines and Circles. Lots of them!

I know I am supposed to ask a specific question, but there's just too many that I would have to ask [it would be like spam] since I missed one week of school because of a family thing and we have an ...
2
votes
1answer
455 views

Trilateration with bounds?

This is a question I posted on Stack Overflow, but I figured you guys would have a better answer for me, so: I'm in need of help solving an issue, the problem came up doing one of my small robot ...
2
votes
3answers
347 views

given two lines in 2D, how to select the angle bisector related to the smallest angle between the lines

I have two lines: first line: $a_1x+b_1y=c_1 \qquad(1)$ second line: $a_2x+b_2y=c_2 \qquad(2)$ I know that the two angle bisectors are expressed by ...
27
votes
9answers
11k views

Is there an equation to describe regular polygons?

For example, the square can be described with the equation $|x| + |y| = 1$. So is there a general equation that can describe a regular polygon (in the 2D Cartesian plane?), given the number of sides ...
5
votes
1answer
316 views

Trying to find an unknown point just with angles

This is my model: What I do know: A, B, C, which form an equilateral triangle Mab, Mbc, Mac which are the middle points Angles x and y, which are the angles formed by the segment from the unknown ...
3
votes
2answers
342 views

Why can any affine transformaton be constructed from a sequence of rotations, translations, and scalings?

A book on CG says: ... we can construct any affine transformation from a sequence of rotations, translations, and scalings. But I don't know how to prove it. Even in a particular case, I found ...
1
vote
3answers
829 views

Find opposite vertices of a rhombus, given the other 2

I am stuck with this problem. I posted an earlier problem with a square, where rotation with i of 90 degrees was possible. This one is a rhombus, how should I proceed? Given ABCD is a rhombus with ...
5
votes
3answers
241 views

What are a , b and c?

$$y = ax^2 + bx + c$$ which is tangent at the origin with the line $y=x$, It is also tangential with the line $y=2x + 3$. Determine the function! Draw a figure! My main question is this solvable? I ...
6
votes
3answers
895 views

What is the area of the portion of 1/8 of an sphere cut off by two parallel planes?

So the problem that I'm trying to solve is as follows: Assume 1/8 of a sphere with radius $r$ whose center is at the origin (for example the 1/8 which is in $R^{+}$). Now two parallel planes are ...
4
votes
1answer
633 views

How to project the surface of a hypersphere into the full volume of a sphere?

The game I mentioned in "Navigating though the surface of a hypersphere in a computer game" is taking shape in here. The world is a 3-sphere where everything belongs. In Euclidean coordinates, for ...
8
votes
2answers
10k views

Finding the intersecting points on two circles

Given 2 circles on a plane, how do you calculate the intersecting points? In this example I can do the calculation using the equilateral triangles that are described by the intersection and centres ...
9
votes
1answer
244 views

Cube skeleton bindings

Imagine that you have a cube skeleton, like so: Further imagine that you have three rubber bands that you can loop through any of the faces. However, only one rubber band may go through any ...
1
vote
1answer
771 views

Find the equation of the plane, in $\bf{r}\cdot n = d$ form

I'll mark the vectors in bold. $p_1 = \bf{i} - 2 \bf{j} + \bf{k}$ $p_2 = 2 \bf{i} + \bf{j} - \bf{k}$ $p_3 = \bf{i} + \bf{j} + \bf{k}$ Could someone please explain to me the way of finding the ...
2
votes
2answers
910 views

Least squares intersection of three circles

Given is a triangle in the plane, with the coordinates of all three vertices known. I need to determine the location of a point $X$, for which the distances to all three triangle vertices are given ...
2
votes
3answers
1k views

Simple gradient/line intersect question

Very, very basic question here: Given an x,y coordinate and a gradient (but no equation), how can I find the x and y axis intercepts? (assuming the line is linear)
10
votes
3answers
11k views

Equation of angle bisector, given the equations of two lines in 2D

I have two lines in 2D expressed with general equation (or implicit equation): First line: $a_1x+b_1y=c_1 \qquad(1)$ Second line: $a_2x+b_2y=c_2 \qquad(2)$ If the two lines are intersecting I will ...