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

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0
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1answer
139 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
votes
1answer
250 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 ...
4
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1answer
148 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 ...
1
vote
1answer
355 views

Find minimum bounding rectangle of an arc

How do you find the minimum bounding rectangle of a circular arc ? You are given the starting point, ending point and another point on the arc. With these points, I've found out the co-ordinates for ...
1
vote
1answer
107 views

Converting an angle in Euclid proof-wise into a relation on 'Cartesian' polynomials

In Is it possible to solve any Euclidean geometry problem using a computer? I claimed that one can convert the statement of a theorem in Euclid into multivariate polynomials such that Groebner basis ...
2
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2answers
506 views

Equation for getting the length of the minor axis (of an ellipse)

I'm looking for an equation that can help me determine the length of the minor axis. I know the length of the major axis and have the Cartesian coordinates of a point somewhere on the ellipse. How ...
4
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2answers
1k views

Find equation for hyperbola

Just taking (failing) a simple algebra class, can't figure this one out and no one can explain it to me and the book just tells me to do it. Find an equation for the hyperbola described: foci ...
5
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2answers
588 views

Find equation of quadratic when given tangents?

I know the equations of 4 lines which are tangents to a quadratic: $y=2x-10$ $y=x-4$ $y=-x-4$ $y=-2x-10$ If I know that all of these equations are tangents, how do I find the equation of the ...
1
vote
1answer
181 views

Cylinder silhouette

I have a parametric representation of a cylindrical shape (well, it's like a cone, but its spike is trimmed). I would like to have an analytic expression for its silhouette lines in terms of the ...
3
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2answers
979 views

Rigid Motions - The product of two rotations around different points is equal to a rotation around a third point or a translation

I'm having some difficulty wrapping my head around rigid motions in a plane. In particular, I'm trying to solve this following problem: In a Euclidean plane, show that the product of two rotations ...
2
votes
2answers
943 views

Finding the z value on a plane with x,y values

so I have the x,y,z value for 3 points to define a plane in 3d space. I need to find the z value of an arbitrary point given the x,y. I can sort of see some ways to calculate this, but they seem ...
11
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1answer
187 views

algebraic versus analytic line bundles

If one has a quasiprojective complex variety X, there is a natural map from the algebraic Picard group to the analytic Picard group. Is this map either injective or surjective? I assume the latter ...
2
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1answer
415 views

Where can I find good resources to practice quadric surfaces and conics?

I need to brush up my knowledge about quadric surfaces and conics. I find myself understanding things better when I work with problems. Do you know any good textbook or website with problems about ...
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2answers
177 views

dividing samples in equal slabs

I have sample data like in above format along X, Y axis. Now what i would like to do is to devide it in "n" number of slabs having fixed values. Now how do i achive this in mathematics(statistics). ...
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3answers
1k views

Formula to Move the object in Circular Path

I want to move one object (dot) in circular path. By using x and y position of that object. Thanks.
1
vote
1answer
69 views

Calculating gradient of a line: how do you know which way to order the points?

Very simple question but I keep getting this wrong! If you have two points e.g. A$(13, 6$) & B$(11, 12)$, Using the gradient formula $m = (y_2 - y_1)/(x_2 - x_1)$ how do you know which of A or B ...
5
votes
4answers
2k views

Why do rhombus diagonals intersect at right angles?

I've looked all over and I can't find a good proof of why the diagonals of a rhombus should intersect at right angles. I can intuitively see its true, just by drawing rhombuses, but I'm trying to ...
2
votes
3answers
246 views

How to obtain Line Equation of the form $ax + by + c = 0$

I'm trying to check if a line hits a rectangle, and for that, I found this nice solution: Line triangle intersection The problem is that, having forgot almost all I ever knew about math, I don't ...
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2answers
1k views

Converting parametric equation to Cartesian one

A curve has parametric equations $$\begin{array}{rcl} x & = & \cos(t)\\\ y & = & \frac{1}{2}\sin(2t) \end{array}\qquad \text{ for }0\leq t \leq 2\pi.$$ Show that the ...
9
votes
1answer
377 views

Navigating though the surface of a hypersphere in a computer game

People in StackOverflow seems not so into this theme, so I thought I could have better luck in here. I had the idea of an spaceship game where the world is confined in the surface of an 4-D ...
15
votes
1answer
625 views

Bath towel on the rope

This question is related to my bath towel, which I hang on a rope, so let's have fun (you can use your own towel to do this experiment in bath-o). There is this rectangle with sides $a<b$. The ...
3
votes
2answers
872 views

Calculation of the touching point of two circles given limited data

I am trying to calculate the touching point of two circles. I have the following information. Circle $1$: Centre $(h,k)$ Radius $r_1$ Circle $2$: Point on Circumference $(x_1,y_1)$ Radius ...
5
votes
4answers
1k views

How to find the distance between a point and line joining two points on a sphere?

How do I calculate the distance between the line joining the two points on a spherical surface and another point on same surface? I have illustrated my problem in the image below. In the above ...
2
votes
1answer
2k views

Getting the third point from two points on one line

My question is the following how can i get the x3,y3 point from x1,y1 and x2,y2 points? the x3,y3 point distance from x1,y1 is 300.
2
votes
1answer
97 views

What does it mean for a sequence of self-homeomorphism of $\mathbb{R}^n$ to converge to a point?

Let $\{f_j\}$ be a family of self-homeomorphisms of $\overline{\mathbb{R}}^n$ and $x,y\in\overline{\mathbb{R}}^n$, where $\overline{\mathbb{R}}^n$ is the one-point compactification of $\mathbb{R}^n$. ...
4
votes
3answers
461 views

The elementary coordinate geometry of polynomials? Of rational expressions? Of radicals?

With a few colleagues, we're trying to design an (intermediate) algebra course (US terminology) where we stress the interplay between algebra and geometry. The algebraic topics we would like to cover ...
2
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1answer
145 views

Complex Square Root

I am not sure where to begin on this: Determine the images of all conic sections with a focus at the origin under the principal branch of the complex square root. I probably have to use the formula ...
4
votes
1answer
329 views

Drop a ball into a bowl in the shape of a paraboloid. How far from the bottom of the bowl will the ball come to rest?

This is a question I like to bring up in a multivariable calculus class from time to time: A sphere of radius 4 is dropped into a bowl-shaped paraboloid given by $z = x^2+y^2$. How close will the ...
2
votes
1answer
124 views

What is the locus such that any vector from it has a given dot product with the given vector?

Consider a given vector $a$ and scalar $d$. What is the set $X$ such that for any $x \in X$ their dot product equals $d$ : $\forall x \in X: x \cdot a = d$ ?
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vote
1answer
375 views

Equation for calculating the volume of liquid in an ellipsoid

I am looking for an equation to calculate the volume of liquid in an underground tank based on a depth reading. The tank is an ellipsoid shape with the following dimensions: ...
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5answers
5k views

Horizontal tank with hemispherical ends depth to capacity calculation

I am trying to find an accurate way of calculating the capacity of an underground tank at a given depth. The tank manufacturer has provided a strapping table for the tank which tells me the capacity ...
-1
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2answers
465 views

How do I add Vectors in Vector Calculus?

For example: Suppose vector u = (-2,3) and vector v = (-5,3) then: (u + v = ?) and (u - v = ?) and (v - u = ?) and (6u = ?) and (-1/8v = ?) and (3u - 4v = ?)
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2answers
6k views

Find an equation of the sphere that passes through the origin and whose center is (5, 10, -9). Help with Calculus III

I have never seen a problem like this before, so I was wondering if anyone could give me help getting started. I'm studying for a quiz on Wednesday. Find an equation of the sphere that passes through ...
4
votes
2answers
1k views

Quadratic equation of an ellipse and ellipse description

I found on Mathworld ( http://mathworld.wolfram.com/Ellipse.html ) that the quadratic equation: $$ax^2 + 2bxy + cy^2 + 2dx + 2fy + g = 0$$ represent an ellipse only when, after defining: $$\Delta = ...
8
votes
4answers
2k views

What Does Homogenisation Of An Equation Actually Mean?

For example, if we have a conic; ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 What does homogenising this equation with another line (say ax + by + c = 0 ) actually mean? As in, what are the graphical ...
9
votes
2answers
2k views

arc-arc intersection, arcs specified by endpoints and height

I need to compute the intersection(s) between two circular arcs. Each arc is specified by its endpoints and their height. The height is the perpendicular distance from the chord connecting the ...
1
vote
2answers
368 views

finding hyperbola asymptotes

Given implicit function F(x,y) = 0, how can I find its asymptotes? EDIT: Sorry, my calculations were wrong. Here is correct function: $F(x,y)=\sqrt{(x-a)^2 + (y-b)^2} - \sqrt{(x-c)^2 + (y-d)^2} - ...
3
votes
4answers
296 views

Equation for a circle

I'm reading a book about Calculus on my own and am stuck at a problem, the problem is There are two circles of radius 2 that have centers on the line x = 1 and pass through the origin. Find their ...
3
votes
2answers
1k views

Average projected area of an ellipsoid

Consider an ellipsoid of semi-axes a, b, c (possibly prolate, b=c). I am interested in the "shadow" of this solid onto a distant plane, in a given direction d=(k,l,m) orthogonal to that plane. By ...
4
votes
1answer
351 views

Is a general (non-homogeneous) quadratic equation in $\mathbb{R}^3$ an ellipsoid?

This sounds like a simple problem, but I can't get it done. Given the general equation $ax^2 + by^2 + cz^2 + dxy + exz + fyz + gx + hy + iz + j = 0,$ what are the requirements on the coefficients so ...
1
vote
1answer
483 views

Locus of osculation of concentric ellipses (elliptic pond ripples)

If you dropped two rocks in a pond, the concentric circles emanating from the two spots would osculate $\infty$ times. The locus of osculating points would form a line. Now imagine that instead of ...
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1answer
259 views

How close are star-convex sets to convex sets?

What interesting properties of convex sets are retained by star-convex sets?
0
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2answers
7k views

How to find shortest distance between two skew lines in 3D?

If given 2 lines $\alpha$ and $\beta$, that are created by 2 points: A and B 2 plane intersection I want to find shortest distance between them. $$\left\{\begin{array}{c} P_1=x_1X+y_1Y+z_1Z+C=0 ...
0
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2answers
3k views

Find the coordinates of a point equidistant from two given points

The Coordinates of two points are A(-2,6) and B(9,3). Find the coordinates of the point C on the x-axis such that AC = BC
3
votes
1answer
260 views

Curve of a fixed point of a conic compelled to pass through 2 points

Suppose that in the plane a given conic curve is compelled to pass through two fixed points of that plane. What are the curves covered by a fixed point of the conic, its center (for an ellipse), its ...
5
votes
2answers
713 views

Tractrix-like curves

Is there a common name for curves, obtained from dragging a point along another curve, similar to how tractrix is obtained by dragging a point along a line? What is a parametric equation of such ...
0
votes
1answer
381 views

how to calculate an arc center

how to find the center of an arc given start and end points
4
votes
2answers
1k views

Argument of the sum of two complex numbers

Let $r$, $s$ be positive real numbers and $\theta$, $\phi$ real numbers with $|\theta -\phi|<\pi$. Then an argument of $re^{i\theta}+se^{i\phi}$ lies between $\theta$ and $\phi$. Can someone give ...
2
votes
3answers
792 views

Nice geometric parallelepiped proof?

Question 1: The volume of a parallelepiped in $\mathbb{R}^n$ with n sides given by the vectors $(x_{1_1}, x_{1_2} ... x_{1_n}), (x_{2_1}, x_{2_2} ... x_{2_n}) ... (x_{n_1}, x_{n_2} ... x_{n_n})$ and ...
1
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2answers
309 views

Equation of a mirror

So let's say you have a curved mirror, $y=y(x)$ with this property: Whenever a ray of light emanates from the origin, it reflects parallel to the x axis. Find the equation of the mirror.