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

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1answer
270 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 ...
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1answer
542 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 ...
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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 ...
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2answers
2k views

rotation by 180 angle

in general i know that if we rotate (x,y) about origin by the 180 degree we will get new image (-x,-y),but suppose that we make rotation not about origin but some other point (a,b) does your result ...
2
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1answer
100 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 point ...
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2answers
957 views

Finding the intersection of a 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 line ...
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2answers
4k 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 ...
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3answers
199 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). ...
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5answers
12k 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 ...
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2answers
249 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 ...
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1answer
236 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 ...
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2answers
196 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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2answers
894 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 ...
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2answers
167 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 ...
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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 ...
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3answers
841 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 ...
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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 $t_2$ ...
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1answer
185 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 of ...
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1answer
147 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
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1answer
370 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 ...
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1answer
443 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, ...
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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 ...
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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 ...
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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) ...
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3answers
420 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
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1answer
432 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 ...
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3answers
320 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 ...
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8answers
9k 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
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1answer
288 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 ...
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2answers
320 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 ...
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3answers
773 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
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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
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3answers
830 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
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1answer
594 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 ...
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2answers
9k 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
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1answer
232 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 ...
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1answer
764 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
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2answers
880 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
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3answers
912 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)
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3answers
10k 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 ...
2
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2answers
557 views

Algebraically determine if lines intersect

I have a programming problem that involves determining if any 4 line segments intersect. (I am testing to see if four points [in a specific order] comprise a quadrilateral). Mathematically speaking, ...
2
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1answer
515 views

analytical geometry rotation

If I have a given point $A(x_a,y_a,z_a)$ in the horizontal plane $z=1400$ and I rotate the plane in such a way that it is perpendicular to a line that makes an angle of $60^{\circ}$ with $Oz$ axes and ...
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2answers
108 views

Is this lemma about the minimal distance of two lines true?

In school, I recently proved a solid geometry excercise by assuming that the following lemma is true: If two lines $g$ and $h$ in the euclidian space are not parallel, and if the lines seem ...
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1answer
156 views

How does Rolle's theorem apply here?

The derivation below was taken from a book on Classical Differential Geometry. It uses Rolle's theorem to find the characteristic line of a family of planes, but I don't see how it applies. Given is ...
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1answer
65 views

Can any smooth planar curve which is closed, be a base for a 3 dimensional cone?

A cone in 3 dimensions has a vertex and a base. The contour of the base is a circle which is a smooth closed planar curve. Can there be a more general cone which can have any smooth closed planar ...
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1answer
232 views

Is there a two dimensional surface like a cone but whose base is elliptic or any non circular but smooth closed curve?

Is there a two dimensional surface like a cone but whose base is elliptic or any non circular but smooth closed curve ? The surface should be smooth everywhere except at the vertex.
0
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1answer
785 views

find the angles of intersection when the line through the points (3,4) and (-5,0) intersects the line through (0,0) and (-5,0)

Find the angle of intersection when the line through the points $(3,4)$ and $(-5,0)$ intersects the line through $(0,0)$ and $(-5,0)$.
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1answer
151 views

perpendicular distance to center of square from line in terms of slope

I am trying to find the relationship between the vertical distance (V) from the center of the square to the line to the perpendicular distance (P) from the center of the square to the line in terms of ...
3
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1answer
286 views

References for the basic theory of surfaces of revolution, cylinders and cones

I'm looking for references to books were the following types of problems about finding the equation defining a surface of revolution, a cylinder or a cone are treated. These are problems that are ...
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1answer
169 views

Hausdorff distance vs. distance of the boundaries

I'm tagging this question homework because I'm more interested in hints than in complete solutions. First let us give a definition. Definition Let $X$ be a metric space. For all $F \subset X, \rho ...