For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, angles.

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5answers
591 views

What is a parallel line?

We are learning vectors in class and I have a question about parallel lines and coincident lines. According to wikipedia a parallel line is: Two lines in a plane that do not intersect or touch at ...
5
votes
1answer
351 views

Ellipse Question

I have only worked with ellipses aligned with the x or y axis. However, how can I approach the following: Suppose we have an ellipse centered at the origin of the following form $$ax^2 + b xy +c y^2 ...
5
votes
2answers
161 views

Warp-like pattern in a closed curve

Given a closed curve in 2D space that intersects itself (transversally, and there's no point in which three paths or more meet), is it possible to look at it as a Celtic knot so when you follow it, ...
4
votes
3answers
280 views

What does the secant value represent?

What does the secant value represent? I know that $$\sec = 1/\cos(\theta)$$ but really I do not know what this value represents, so I need your help. A clear example with images would be appreciated. ...
4
votes
1answer
87 views

The curve has constant torsion.

Question: Show that when the curve $c_1=c_1(t)$ has constant torsion $\tau$, the curve $$c_2=c_2(t)=-\frac{1}{\tau}N+\int_{t_0}^{t}B(u)du$$ has constant curvature $-\tau$ or $+\tau$. What I ...
4
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3answers
276 views

There isn't a product operation that is commmutative on $ \mathbb{R}^{n} $ that satisfies all the field axioms for $ n \geq 3 $.

This proof is broken down into simple easy algebra and vector questions. I would like to discuss different answers and approaches. Please see pg 162-163 on books.google.ca/books?isbn=0387290524 ...
4
votes
3answers
870 views

Determinants and volume of parallelotopes

The absolute value of a 2 by 2 matrix determinant is the area of a corresponding parallelogram with the 2 row vectors as sides. The absolute value of a 3 by 3 matrix determinant is the volume of a ...
3
votes
2answers
93 views

Area of a triangle in terms of areas of certain subtriangles

In triangle $ABC$ , $X$ and $Y$ are points on sides $AC$ and $BC$ respectively . If $Z$ is on the segment $XY$ such that $\frac{AX}{XC} = \frac{CY}{YB} = \frac{XZ}{ZY}$ , then how to prove that the ...
3
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2answers
146 views

Check Points are line, triangle, circle or rectangle

How to determine geometric properties of four distinct points in a plane (x1,y1), (x2,y2), (x3,y3), (x4,y4) represented in the 2-D Cartesian coordinate system, whether these four points are on a ...
3
votes
1answer
1k views

Calculate Camera Pitch & Yaw To Face Point

How do you calculate pitch & yaw for a camera so that it faces a certain 3D point? Variables Camera X, Y, Z Point X, Y, Z Current Half Solution Currently I know how to calculate the pitch, ...
3
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4answers
3k views

Deriving the Area of a Sector of an Ellipse

A sector $P_1OP_2$ of an ellipse is given by angles $\theta_1$ and $\theta_2$. Could you please explain me how to find the area of a sector of an ellipse?
3
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1answer
182 views

Triangle and Incircle

Today in class me and my friend were discussing cool problems that we've done. And he asked me to.find with proof something interesting. Triangle ABC has right angle at B and we drop a perpindicular ...
3
votes
2answers
244 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 ...
3
votes
6answers
16k views

Showing three points are collinear

Show that (-1,8), (1, -2) and (2,1) lie on a common line. Any help understanding how to go about doing this is greatly appreciated.
3
votes
3answers
1k views

A parallelogram and a line joining a vertex to the midpoint of opposite side

In a parallelogram ABCD. M is the midpoint of CD. Line BM intersects AC at L and it also intersects AD extended at E. Prove that EL=2BL PS: This is not a homework problem. I was solving geometry for ...
3
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3answers
3k views

How to find where 3 lines intersect

I've got a programming exercise I need to do, but I just can't figure out the math part. I need to check if 3 of 6 lines intersect in the same point. I am given the equation ax+by=c, and I input ...
2
votes
1answer
141 views

Why are two definitions of ellipses equivalent?

In classical geometry an ellipse is usually defined as the locus of points in the plane such that the distances from each point to the two foci have a given sum. When we speak of an ellipse ...
2
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1answer
2k views

Calculating circle radius from two points on circumference (for game movement)

I'm designing a game where objects have to move along a series of waypoints. The object has a speed and a maximum turn rate. When moving between points p1 and p2 it will move in a circular curve ...
2
votes
3answers
937 views

3D to 2D rotation matrix

I have been trawling through this forum but am struggling to understand the maths a bit. Currently I have a 2D plane within a 3D space and I have the coordinates for them. I want to work on this 2D ...
2
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0answers
85 views

Generalizations of equi-oscillation criterion

When constructing minimax (sup-norm) polynomial approximations of real-valued functions, well-known results say (roughly speaking) that optimal solutions are characterized by the fact that they have ...
2
votes
2answers
2k views

Counting number of distinct regions with intersecting circles

Given $n$ circles of possibly different radii, how many distinct regions can there be? For small $n$, I can work it out with pictures. (I'm pretty sure $n=4$ can yield 13 distinct regions, but not ...
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vote
2answers
39 views

How to to a better approach for this :?

If, $$x\cos A+y\sin A=k=x\cos B+y\sin B$$ Then find $(\cos A)(\cos B)$, $(\sin A)(\sin B)$ and $\cos A+\cos B$ and express them in terms of $x,y,k$ I found a solution but it included a really ...
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2answers
42 views

Geometrical place with circles…

How to find the geometrical place of all centers of a circles that tangent from inside to the circle $x^2+y^2=R^2$ and the $y$-axis? (Suppose that $x,y\geq 0$)
1
vote
1answer
76 views

Diophantine quartic equation in four variables, part deux

A recent Question asked for all positive integer solutions of a simple quartic in four unknowns: $$ wxyz = (w+x+y+z)^2 \tag{1}$$ whose satisfaction is necessary for the integer side lengths ...
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vote
2answers
653 views

How to find the intersection of three spheres (full solutions)?

The three equations of spheres are given $(x-x_{1})^2+(y-y_{1})^2+(z-z_{1})^2=a^2$ $(x-x_{2})^2+(y-y_{2})^2+(z-z_{2})^2=b^2$ $(x-x_{3})^2+(y-y_{3})^2+(z-z_{3})^2=c^2$ How do I find $(x,y,z)$ ...
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vote
1answer
3k views

Converting Lat/Long coords to Cartesian X/Y, then calculating shortest distance between point & line segment

I'm having an issue with accuracy when converting Lat/Long coordinates to X,Y and then finding the shortest distance from a Point to a Line with said coordinates. The distance is off by around 40-50% ...
1
vote
1answer
317 views

Sum of Angles in a Triangle.

Can anyone please explain how to form a better idea in understanding Sum of measures of angles in a triangle are 180 degrees.
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4answers
84 views

Equal perimeter and area

Find all triangles of which perimeter and area are numerically equal. I have got solution for right angle triangles but not of others
0
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1answer
50 views

What is the angle $<(BDE,ADH)$?

What are ways to determine the angle $<(BDE,ADH)$ (the angle between the two planes passing through the points B,D,E and the points A,D,H respectively)?
0
votes
1answer
78 views

Area of quadrangle

In the quadrangle $ABCD$, the points $E,F,G,H$ are the midpoints of respectively $AB, BC, CD, DA$. We know that area $\triangle AHL=a$, $\triangle DIG=b$, $\triangle FJC=c$, $\triangle EBK=d$. Prove ...
0
votes
1answer
100 views

A curve such that all lines on the plane intersect it : cont..

Further to this question (which appears more or less settled); "Is there a curve on plane such that any line on the plane meets it (a non zero ) finite times ?" I ask now the upper bounds of the ...
0
votes
1answer
216 views

$n$ Lines in the Plane

How am I to "[u]se induction to show that $n$ straight lines in the plane divide the plane into $\frac{n^2+n+2}{2}$ regions"? It is assumed here that no two lines are parallel and that no three lines ...
0
votes
1answer
71 views

Square coloring

There are 3 red axis-aligned interior-disjoint squares. There are 3 blue axis-aligned interior-disjoint squares. Is it always possible to find a pair of 1 red square and 1 blue square, such that ...
0
votes
3answers
2k views

How do you find the distance from a point to a plane?

I am having trouble with this: Find the distance from the point $(1,1,1)$ to the plane $2x+2y+z=0$. Any ideas? Thanks.
0
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1answer
1k views

How can I find equivalent Euler angles?

I have a rotation over time represented as a series of Euler angles (heading, pitch and bank). I'd like to represent the individual rotation curves as continuously as possible. An anyone help me ...
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votes
2answers
526 views

In △ ABC, D is the midpoint of AB, while E lies on BC satisfying BE = 2EC. If m∠ADC=m∠BAE, what is the measure of ∠BAC in degrees?

In △ABC, D is the midpoint of AB, while E lies on BC satisfying BE = 2EC. If m∠ADC=m∠BAE, what is the measure of ∠BAC in degrees? I know already that angle A and angle D are congruent because ...
8
votes
1answer
272 views

How to parameterize an orange peel

I'm trying to parametrize the space curve determined by the boundary of a standard orange peel: for example, the one on this photo: For example, the ideal curve would be inside the unit cube; have ...
7
votes
1answer
164 views

Rotation of $\mathbb{R}^3$ by using quaternion

Express the rotation of $\mathbb{R}^3$ by $\frac{\pi}{3}$ about the $x=y=z$ axis by using quaternions and identifying $\mathbb{R}^3$ with $(i,j,k)$-space. Thoughts: From my point of view, every ...
7
votes
3answers
7k 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 ...
6
votes
1answer
139 views

Trajectories on the $k$-dimensional torus

Let $r_1,\dots,r_k$ be irrational and linearly independent over $\mathbb Q$. My intuition clearly tells me that the set $$\{(nr_1,\dots,nr_k)+\mathbb Z^k:n\in\mathbb N\}$$ is dense in $\mathbb ...
5
votes
4answers
283 views

How find this maximum $S_{\Delta ABC}$

in $\Delta ABC$,and $\angle ABC=60$,such that $PA=10,PB=6,PC=7$, find the maximum $S_{\Delta ABC}$. My try:let $AB=c,BC=a,AC=b$, then $$b^2=a^2+c^2-2ac\cos{\angle ABC}=a^2+c^2-2ac$$ then ...
5
votes
2answers
646 views

Aren't asteroids contradicting Euler's rotation theorem?

I am totally confused about Euler's rotation theorem. Normally I would think that an asteroid could rotate around two axes simultaneously. But Euler's rotation theorem states that: In geometry, ...
5
votes
1answer
204 views

For a polygon on complex plane, when are the vertex 'Fourier coefficients' non-zero

Consider an $n$-sided convex polygon $P$ that contains the origin in the complex plane. Let the $j$-th vertex be denoted $z_j = r_j e^{i\theta_j}$ ($0 \leq \theta_j < 2 \pi$) for $j= 1 \dots n$. ...
5
votes
2answers
613 views

What is the name for a shape that is like a capsule, but with two different radii?

I'm looking for the name of a shape that is like a capsule, but where each circle can have different radii. The shape could be described using two circles (two centers and two radii). Something like ...
4
votes
1answer
177 views

Maximal volume for given surface area of an $n$-hedron

Is there a term for a polyhedron with $n$ faces (or, similarly, $n$ vertices) that maximises the enclosed volume for a given surface area (equivalently, minimises the surface area for a given volume)? ...
4
votes
1answer
243 views

Shortest path on unit sphere under $\|\cdot\|_\infty$

Let $X$ be $\mathbb{R}^3$ with the sup norm $\|\cdot\|_{\infty}$. Let $Y=\{x\in X: \|x\|_{\infty}=1\}$. For $x,y\in Y$ define $d(x,y)$ to be the arc length of shortest paths on $Y$ joining $x,y$. (It ...
4
votes
3answers
397 views

Tetrahedral torus

Is it possible to form a closed loop by joining regular (platonic) tetrahedrons together side-to-side, with each tetrahedron having two neighbours? It should be a loop with a hole in, as can be done ...
4
votes
3answers
1k views

Calculating an Angle from $2$ points in space

Given two points, around an origin $(0,0)$, in $2$D space how would you calculate an angle from $p_1$ to $p_2$. How would this change in $3$D space?
3
votes
2answers
64 views

Drawing a triangle from medians

Is it possible to draw a triangle, if the length of its medians $(m_1, m_2, m_3)$ are given only? Someone asked me this question, but I can not see it. Is it really possible? UPDATE Apart from the ...
3
votes
1answer
76 views

If $\gamma$ is spherical, then the equation $\frac{\tau}{\kappa}=\frac{d}{ds}(\frac{\dot{\kappa}}{\tau \kappa^2})$ holds.

Question: Let $\gamma (t)$ be a unit-speed curve with $\kappa(t)\gt0$ and $\tau(t)\neq0$ for all $t$. Show that, if $\gamma$ is spherical, i.e., if it lies on the surface of a sphere, then ...