Geometry Definition and 999 Threads

Can anyone derive the distance formula of a hyperbola for me, please? I have not found the derivation on the internet. I can't get any clue from the picture of hyperbola.

Summary:: We have a rotating arm, offset from the centre of rotation by a certain length, which is controlled by varying the length of a control rod. Need the angle of the rotating arm in terms of length of the rod. The blue line is a fixed column structure. CE and BD form the rotational...

So far all I can work out is that the angle of incidence of the outer two and inner two rays is zero degrees, however, I can't work out how to get started on the problem. I feel like I need to use vertical slowness rather than the normal snell's law since I'm working with a dZ rather than a dX...

My partner asked me about questions no. 8 and 9. Number 8 asks about what is the area of the quadrilateral. Number 9 asks about what number is below the number 25. Those are questions for Elementary School Math Olympiads in my country but both of us were having a hard time figuring them out...

Summary:: Suppose that [x, y] = e^{-3t} [-2, -1] is a solution to the system $x' = Ax$, where A is a matrix with constant entries. Which of the following must be true? a. -3 is an eigenvalue of A. b. [4, 2] is an eigenvector of A. c. The trajectory of this solution in the phase plane with axes...

Hi I have noticed that while I have the grasp of the theoretical underpinnings of linear algebra, I need work on applying it to geometric problems (think computer vision and rigid body motion). So, I am looking for a book that allows me to practice 3D geometry problems. Is there any obvious...

I am uncertain if this belongs in the differential geometry thread because I don't know what area of mathematics my question belongs into begin with, but of the math threads on physics forums, this one seems like the most likely to be relevant. I recently watched a video by PBS infinite series...

I'm trying to find angles α and β. No additional information except: d, h, a. I already tried to figure it out by using isosceles triangles, but this is only true when there is a equilibrium of forces. I thought there are similar triangles incorporated, but I get too many unknown variables...

Hello. If a closed ball is expanding in time would we say it's expanding or dilating in Riemannian geometry? better saying is I don't know which is which? and what is the function that explains the changes of coordinates of an arbitrary point on the sphere of the ball?

Hello. I am confused with this matter that how should we write the transformation matrix for an expanding space. consider a spacetime that is expading with a constant rate of a. now normally we scale the coordinates for expansion which makes the transformation matrix like this: \begin{pmatrix}...

Before looking at the proof of basic theorems in Euclidean plane geometry, I feel that I should draw pictures or use other physical objects to have some idea why the theorem must be true. After all, I should not just plainly play the "game of logic". And, it is from such observations in real...

Every explanation about scaling a 2D vector, or equivalently having a line segment PQ on cartesian plane and then find a point R on the line PQ satisfying PR/PQ = r (fixed given r) starts with that one specific case in the picture. A formula for the coordinates of R is then given for that case...

I am going over review to cover some trig fundamentals. Am stuck and am looking for the right place to get help for a question.

Is there ever an instance in differential geometry where two different metric tensors describing two completely different spaces manifolds can be used together in one meaningful equation or relation?

Point D divides side AC, of triangle ABC, so that |AD|: |DC| = 1:2. Prove that vectors \vec{BD} = 2/3 \vec{BA} + 1/3 \vec{BC}.

Hi, I just finished up with Riemann Geometry not to long ago, and did something with complex geometry on kahler manifolds. In your opinion what would be a next logical step for someone to study? I am very interested in manifold theory and differential geometry in general. I'm somewhat familiar...

Puzzles by Catriona Shearer. The Tilted Twin (and other delights)

Hello All, I have yet another MCNP question. I received the following error "geometry error: no intersection found mcnp" when trying to run a a simulation. I looked at the output and according to it I have an infinite volume in cells 14 and 500. I plotted the geometry and don't see how its...

Hello everyone! I have been looking for a general equation for any regular polygon and I have arrived at this equation: $$\frac{nx^{2}}{4}tan(90-\frac{180}{n})$$ Where x is the side length and n the number of sides. So I thought to myself "if the number of sides is increased as to almost look...

The metric for 2-sphere is $$ds^2 = dr^2 + R^2sin(r/R)d\theta^2$$ Is there an equation to describe the area of an triangle by using metric. Note: I know the formulation by using the angles but I am asking for an equation by using only the metric.

Thinking of triangular pattern

https://en.wikipedia.org/wiki/Flatness_problem The flatness problem (also known as the oldness problem) is a cosmological fine-tuning problem within the Big Bang model of the universe. The fine-tuning problem of the last century was solved by introducing the theory of inflation which flattens...

Hi, I'm already familiar with differential forms and differential geometry ( I used multiple books on differential geometry and I love the dover book that is written by Guggenheimer. Also used one by an Ian Thorpe), and was wondering if anyone knew a good book on it's applications. Preferably...

Summary: Given three points on a positive definite quadratic line, I need to prove that the middle point is never higher than at least one of the other two. I am struggling to write a proof down for something. It's obvious when looking at it graphically, but I don't know how to write the...

Honestly I don't know where to begin. I started differentiating alpha trying to show that its absolute value is constant, but the equation got complicated and didn't seem right.

I am trying to derive the radial momentum equation in the equatorial Kerr geometry obtained from the equation $$ (P+\rho)u^\nu u^r_{;\nu}+(g^{r\nu}+u^ru^\nu)P_{,r}=0 \qquad $$. Expressing the first term in the equation as $$ (P+\rho)u^\nu u^r_{;\nu}=(P+\rho)u^r u^r_{;r} $$ I obtained the...

AB=AC. P is on ac such that AP=3PC. Q on CB such that CQ=3BQ. Need to find the length of PQ. I know i can use the Cosine theorem, but the answer is without Cosine.

I was able to find the equation of an ellipse where its major axis is shifted and rotated off of the x,y, or z axis. However, I could not find anywhere an equation for a spheroid that does not have its axis or revolution along the x,y, or z axis. How might I go about deriving such an...

I am throwing a bachelor party for my brother, who is currently getting his PhD in Math at columbia, and as you might expect, he is not very much of a party animal. I want to throw him a party he’ll enjoy, so I came up with scavenger hunt in the woods, where every step in the scavenger hunt is a...

Reading the interesting book "Groups_and_Manifolds__Lectures_for_Physicists_with_Examples_in_Mathematica", in the introduction it is stated: (...) we have, within our contemporary physical paradigm, a rather simple and universal scheme of interpretation of the Fundamental Interactions and of...

Can someone tell me why the proper time between αβ is not t2+x2 but rather t2-x2? Background:

Hello guys, is it possible to "see" the mean value theorem when one is only thinking of numerical values without visualizing a graph? Perhaps through a real world problem?

Hello all, I need some help on two exercises from Kiselev's geometry, about straight lines. Ex 7: Use a straightedge to draw a line passing through two points given on a sheet of paper. Figure out how to check that the line is really straight. Hint: Flip the straightedge upside down. I would...

I have tried a lot by angle chasing e.g. let ∠ABC=x° then ∠ACB=90°-x°. As AU=AV=radius of circle so ∠AUV=∠AVU=45°. I've connected U,D and V,D. Then ∠UDV=135° etc. But I haven't found any way to get near of proving AE=DE. I have also tried to prove 'the area of triangle AEU= area of triangle...

I can construct examples that are less than or equal to ##4n## quite easily, but for the life of me I cannot come with example where it's greater than

Through symmetry of parallelogram,I have come to this: Here 1,2,3,4 denotes the area of the particular regions.Then I am stuck.Please help what to do next or whether there is any other way.

-I tried to draw the forces on the hoop when it is in the equilibrium state. I know there are friction and normal force on the contact point of the shaft and the hoop -I also put the weight force to the M object -But when i used the torque equilibrium, where the pivot is the contact point of the...

Hi, Guys, I am building light electric vehicle for fun and hobby. I am experimenting with stuff, but have some difficulty calculating a reasonable geometry for the suspension. It is a SINGLE arm suspension and I am planing to use Fox Float Air Shock with length 190mm, it has pressure control...

Homework Statement :[/B] The following expression stands for the two angular bisectors for two lines :\frac{a_{1}x+b_{1}y+c_{1}}{\sqrt{a_{1}^{2}+b_{1}^{2}}}=\pm \frac{a_{2}x+b_{2}y+c_{2}}{\sqrt{a_{2}^{2}+b_{2}^{2}}}\qquad Homework Equations The equations for the two lines are : ##a_1x + b_1y +...

Are the concepts of the rules of Algebra, Geometry and Probability things that all humans have some instinctive grasp at some level, or things that we basically learn from others, therefore cultural? Let me explain. I once saw an experiment with a mommy rat. She had 4 puppies, and someone put a...

Homework Statement The angle alfa has its vertex at a point O, from one of its sides the point M is taken from which the perpendicular to the other side is made with the point N. In the same way, from the other side point K is taken and from there the perpendicular to the other side is traced...

Homework Statement tl;dr: looking for a way to find the intersection of three cones. I'm currently working with a team to build a Compton camera and I've taken up the deadly task of image reconstruction. Background Theory: https://en.wikipedia.org/wiki/Compton_scattering For a single Compton...

Recently I started looking back at some basic mathematical principles, and I started thinking about the theorem of corresponding angles. It's such a basic idea that it seems obvious on an intuitive level, but despite that (or possibly because of that) I can't think of a good way to formally...

Consider a circle with a chordal line dividing the area into two unequal parts. It seems to be accepted practice to call the smaller of these parts a circular segment. Is there a generally accepted name for the larger area? I've been writing some material where this geometry arises, and I've...

So far i have 270/360× (pi)r^ i don't know what to do next please help.

This is my first video!

Im planning on taking a course on classical differential geometry next term. This is the outline: The differential geometry of curves and surfaces in three-dimensional Euclidean space. Mean curvature and Gaussian curvature. Geodesics. Gauss's Theorema Egregium. The textbook is "differential...

Homework Statement Find the coordinates of those points on the lines (x+1)/2 = (y+2)/2 = (z-3)/6 which is at a distance of 3 units from the point (1,-2,3) 2. Relevant methods 1) assume a point, use distance formula- (very calculative) 2) write vector equation of line, find foot of...

Given ##ds^2 = y^{-2}(dx^2 + dy^2)##, I am trying to prove that a demicircle centred on the x-axis, written parametrically as ##x=r\cos\theta + x_0 ## and ##y= r \sin \theta ## are geodesics. Where ##r## is constant and ##\theta \in (0,\pi)## I have already found the general form of the...