Questions on coordinate systems, a geometric method where a point in n-dimensions is assigned a corresponding n-tuple of numbers.

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Relationships between positions on a sphere in a global coordinate system

Imagine I have a camera which is observing an object. The camera can move around the abject, by traversing a virtual sphere, where the radius of the sphere is the distance between the object centroid ...
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0answers
45 views

First person camera rotation (conversion between spherical to cartesian coordinates)

A similar question has been asked before on Stack Overflow but no satisfying answer. Basically I'm following this tutorial, and I can't exactly figure out how the guy deduced these two equations to ...
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0answers
34 views

How to convert Cartesian coordinates to Tetrahedral (Quadriplanar) coordinates.

I'm looking for a method to convert 3D Cartesian coordinates to 4D tetrahedral coordinates. I need to implement that coordinate system in shader to create a procedural texture. I can generate separate ...
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2answers
112 views

All Intersection points of two spheres having arbitary centres?

I have read much about intersection of two spheres from spheres-intersect , circlesphere and collision-points but all are based on the assumption of spheres located at origin or $x$-axis or some ...
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1answer
33 views

How do I find the eccentricity of this inscribed ellipse given one of its focus?

An ellipse is inscribed in Triangle ABC: $A(0,0), B(1,2)$ and $C(2,0)$. One of the focus of the ellipse is at S(1,$\frac {1}{2}$). Find the eccentricity $e$ of this ellipse. I know this looks like an ...
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1answer
67 views

Book on coordinate transformations

I am looking for a book that covers various coordinate systems in 3 dimensions, various methods of representing rotations and other transformations like rotation matrices and quarternions, including ...
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1answer
31 views

Calculate the center point of the highest density of xy points

I have a square grid (200 x 200). I would like users to be able to vote on their next desired position of an object in this space. When a user votes I add the XY position to an array. Currently to ...
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0answers
26 views

Inseparable and non-standard solutions to a differential equation which is also solvable via separation of variables

I have a relatively straightforward differential equation. Actually, the paraxial wave-equation: \begin{eqnarray} \left(\nabla^{2}-2ik\frac{\partial}{\partial z}\right)\psi=0 \end{eqnarray} ...
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0answers
13 views

Finding curvature in cylindrical coordinates

I'm trying to find the formula for curvature in cylindrical coordinates, but I'm not sure what to do. The formula I usually use (https://en.wikipedia.org/wiki/Curvature#Local_expressions) yields a ...
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1answer
37 views

Formalizing differation of a function $f$ by function $g$

Sometimes (usually in physics) i encounter expressions such as $\dfrac{d}{dg(x)} f(x)$ This expression is quite intuitive: what is the change in $f(x)$ with respect to change in $g(x)$? How can i ...
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3answers
54 views

Finding the coordinate C.

A triangle A, B, C has the coordinates: A = (-1, 3) B = (3, 1) C = (x, y) BC is perpendicular to AB. Find the coordinates of C? My attempt: Grad of AB = $(3-1) ÷ (-1-3) = -0.5$ Grad of BC = 2. ...
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0answers
49 views

Coordinate system construction and change for a (X, Y, Z) cloud of points

I’m working on a project in an aerospace company. I have to post process a cloud of points with X Y Z coordinates in Excel spreadsheet. I need to build different coordinate frame (origin and x y z ...
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5answers
150 views

Finding the locus in terms of a and b

Find the locus of the point equidistant from $(a+b,b-a)$ and $(a-b,a+b)$ Okay so I don't know how do I find the locus of the point in terms of $ a$ & $b $.
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1answer
34 views

Tangent at point $P_1$ other than $(0,0)$ on the curve $y =x^3$ meets the curve again at $P_2$…

Problem : Tangent at point $P_1$ other than $(0,0)$ on the curve $y =x^3$ meets the curve again at $P_2$ The tangent at $P_2$ meets the curve again at $P_3$ and so on. If $\frac{\text{area} ...
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0answers
19 views

peculiarity in co-ordinate transformation

I came across a peculiar thing while solving a problem concerning non-inertial frames(rotating frames). I'll try to give the problem formulation as simply as I can. Here I go Consider 3 frames - F1, ...
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0answers
24 views

Parabolic coordinates as functions of $x$, $y$ and $z$

OK, so I have this problem: Express the parabolic coordinates $\textit{u}$, $\textit{v}$ and $\varphi$ as functions of the cartesian coordinates $x$, $y$ and $z$, how does $\textit{u}$, $\textit{v}$ ...
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2answers
114 views

Finding equation of chord of hyperbola.

Equation of chord of hyperbola joining points $(a\sec\phi,b\tan\phi)$ and $(a\sec\phi_1,b\tan\phi_1) $ $$y-b\tan\phi=\frac{b\tan\phi-b\tan\phi_1}{a\sec\phi-asec \phi_1}(x-a\sec\phi) $$ This reduces ...
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0answers
28 views

partial states or partial probability?

I am trying to figure out an alternative way of representing a state probability space, to make certain ideas clearer (that I don't need to discuss here). Let's say I have a system of two elements, A ...
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1answer
46 views

Find the circle circumscribing a triangle related to a parabola [duplicate]

Consider the following lines $x-y-1=0$ $x+y-5=0$ $y=4$ The line 1 is the axis of the parabola, the line 2 is the tangent at the vertex to the same parabola, and the line 3 is another tangent to ...
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1answer
68 views

Is a coordinate system a requirement for a vector space?

In other words, can a vector space exist, and not have a coordinate system? I'm asking because in the definitions that I've seen of a vector space, there's no mention of a coordinate system.
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0answers
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Atan2 - Dealing with non-standard coordinate systems when calculating angles

I am trying to use atan2 to calculate the angle between two points. With atan2, angles increase counterclockwise, and zero is to the right i.e. The angle originates at positive x, increases up into ...
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0answers
10 views

Scale factors and metric in cylindrical and spherical coordinates - isotropy of space [duplicate]

In cylindrical (polar) coordinates, the scale factors are $$h_r=1$$ $$h_{\theta}=r$$ $$h_z=1$$ Would it be correct to say that $h_i$ do not depend on $\theta$ because space is isotropic (has the same ...
2
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0answers
77 views

Scale factors in cylindrical coordinates - geometrical meaning

I am trying to make sense of the scale factors in cylindrical coordinates and their geometrical meaning. To start with something simpler, begin with Cartesian coordinates: $$h_x=h_y=h_z=1$$ One can ...
2
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1answer
84 views

Find the locus of intersection of tangents to an ellipse if the lines joining the points of contact to the centre be perpendicular.

Find the locus of intersection of tangents to an ellipse if the lines joining the points of contact to the centre be perpendicular. Let the equation to the tangent be $$y=mx+\sqrt{a^2m^2+b^2} $$ This ...
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0answers
31 views

Plot ENU points on a map having an origin GPS Lat and Long

I have a GPS coordinate point to denote my origin: 8.6258,47.3325 Then I have a series of ENU (east, north up) points in meters that represent the distance from ...
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2answers
34 views

Distance in coordinate system [closed]

The line $y = \frac{3}{4}x + m$, where $m\ne0$ , intersects the $x$-axis at point $P$ and the $y$-axis at $Q$. Which is larger: Quantity 1 : The distance from $P$ to the origin $( 0 , 0)$ Quantity ...
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0answers
29 views

Rotation matrices and angular velocity

Let us assume I have an object O with axis $x_{O}$, $y_{O}$, $z_{O}$, with different orientation from the global frame S with $x_{S}$, $y_{S}$, $z_{S}$ (I don't care about the position). Now I know ...
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1answer
63 views

What is the cross product in spherical coordinates?

The curl of an arbitrary vector, $\vec{A}$ is The curl of an arbitrary vector $\vec{A}$ in spherical coordinates \begin{align*} \nabla \times \vec{A} &= \frac{1}{r^{2}\sin{\theta}}\left| ...
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3answers
46 views

Why does the volume element in spherical polar coordinates contain a sine of the zenith angle?

The volume element in Cartesian coordinates is $dx dy dz$, the volume of a rectangular prism with side lengths being the length elements along the three rectangular axes. In spherical polar ...
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1answer
22 views

Tensor transformations in curvilinear coordinates

According to tensor notation, to transform a vector from coordinates $x^\mu$ to $x^{\mu'}$, one applies the rule $$ V^{\mu'} = \frac{\partial x^{\mu'}}{\partial x^\mu} V^\mu $$ However, for example ...
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0answers
42 views

Need help to rotate a 3D point around a line and find the coordinates

How do I solve this?: Rotate p=[2,1,3] 60 degrees anti-clockwise around the line from a=[1,-1,1] to b=[3,-3,3]. What is the coordinates of p'? The first thing I've to do is to find vector AB and ...
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3answers
52 views

Find the locus of point R.

A ladder, PQ, 5m long leaning against a wall. Assuming the floor and the wall represent the x-axis and y-axis respectively. R is a point on PQ such that RQ is 2 times RP. Given that P(0, b) and Q(a, ...
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2answers
17 views

how to map series of coordinates onto a series of coordinates with different resolution

I have a set of target coordinates and a set of actually clicked coordinates which should be approximately the same, but not identical. The y coordinates are equal, however, the x-coordinates differ, ...
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0answers
28 views

What is the $x$ and $y$ equation for this mechanism

I have this mechanism: I can rotate a and b using servo motors. My question is this: how is the angle rotation of the motors related to the $X$ and $Y$ position of $c$ ? I mean: I want to be able ...
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1answer
54 views

How I could prove that this vector is a unit vector of a basis?

I need prove that $$ ê_j(P)=\frac{\frac{\partial \vec{r}}{\partial q_i}}{| \frac{\partial \vec{r}}{\partial q_i}|} $$ is a unit vector of another basis, I think I have to apply that the partial of ...
2
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1answer
28 views

Modification of Manhattan Geometry

In a gridworld, the usual metric used is the Manhattan Metric. From a given coordinate, there are only four possible directions: up, down, left, right. I want to modify the metric by giving four ...
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1answer
243 views

Angle between a pair of tangent lines

NB: Everything I know about a pair of straight lines is from this PDF . I have made use of quite a few formulas given in the PDF in my answer. Q: Find the equations of the tangents from the point ...
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0answers
55 views

Convert from Rotation Matrix to Fixed-Axis (Extrinsic) zxz Euler Angles

How can an arbitrary rotation matrix $R = \left(\begin{matrix} R_{11} & R_{12} & R_{13} \\ R_{21} & R_{22} & R_{23} \\ R_{31} & R_{32} & R_{33}\end{matrix}\right)$ be ...
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0answers
56 views

Calculate position of a new point in 3D with respect to a point

I am having a set of points in 3D in a global coordinate system. All points are moving with respect to time. Now I want to freeze a point and want to calculate the relative displacement of other ...
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0answers
56 views

Combining vector coordinates in $\mathbb{R^3}$ from two perspectives

DISCLAIMER: I would like to apologize beforehand for asking this stupid question. Somehow I have a feeling that I should know this, even with basic math. Some of the examples I will give are probably ...
3
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0answers
57 views

On teaching elementary co-ordinate geometry. [closed]

I am presently teaching eleventh grade (XI standard) students an introductory course in co-ordinate geometry with a focus on preparations for competitive exams. I have seen books like S.L.Loney's ...
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1answer
29 views

How to isolate $\eta$ from the oblate spheroidal coordinates?

Background: What I am trying to do is to have a expression to convert rectangular coordinates to oblate spheroidal coordinates (what is called inverse transformation); in the wikipedia article Oblate ...
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1answer
43 views

Why do equations in these forms always intersect the point $(1,1)$?

I was fooling around with a graphing calculator, and I discovered something that surprised me a little. Suppose I have a quadratic equation in the following form, where $n \in \mathbb{R}$: $$y=x^n$$ ...
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1answer
32 views

Change of basis (coordinates)

can somebody help me with this by giving me a hint how to solve these $2$ $?$: Given a point $P$ with coordinates $(10,4)$ in a fixed coordinate system. a new coordinate system with origo in ...
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2answers
48 views

how to compute length in polar coordinates?

The line element $\Delta s^2$ is suppose to be an invariant of Euclidean space. In standard coordinates $\Delta s^2=\Delta x^2+\Delta y^2$ while in polar coordinates $\Delta s^2=\Delta r^2+r^2\Delta ...
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0answers
22 views

Short way to find the grad, curl and div in curvilinear coordiantes?

In orthogonal curvilinear coordinates $(\zeta_1, \zeta_2, \zeta_3)$ with scale factors $h_1, h_2, h_3$ the grad, curl and div can be written as: $$\nabla (\psi)=\frac{\hat e_1}{h_1} \frac{\partial ...
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1answer
31 views

How to find nature of algebraic graph? [closed]

How to find nature of any algebraic graph when graph points are given? Ex : $x = (1,2,3,4)$ $y= (3,8,15,24)$ What is the nature of graph ? P.S. I want to know short-cuts for all possible graphs?
2
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3answers
90 views

What are the dual basis vectors?

What exactly are dual basis vectors such as those which arise in non-orthogonal co-ordinate systems? What is their physical interpretation. Please note, I don't know much tensor calculus yet. I am ...
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1answer
45 views

Find pair of tangents to circle

Find pair of tangents from $(3,4)$ to circle $x^{2}+y^{2}=9$ We have to find tangents to circle. I have supposed a line $$y = mx + c.$$ By condition of tangency I have $$c = 3\sqrt{1+m^{2}}.$$ ...
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1answer
24 views

Integer points on a line.

Is there a formula for finding the integral points on a line segment? I've tried moving through the graph to find out the answer.