Geometry (from the Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a geometer.
Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Gauss' Theorema Egregium ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in an Euclidean space. This implies that surfaces can be studied intrinsically, that is as stand alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry.
Later in the 19th century, it appeared that geometries without the parallel postulate (non-Euclidean geometries) can be developed without introducing any contradiction. The geometry that underlies general relativity is a famous application of non-Euclidean geometry.
Since then, the scope of geometry has been greatly expanded, and the field has been split in many subfields that depend on the underlying methods—differential geometry, algebraic geometry, computational geometry, algebraic topology, discrete geometry (also known as combinatorial geometry), etc.—or on the properties of Euclidean spaces that are disregarded—projective geometry that consider only alignment of points but not distance and parallelism, affine geometry that omits the concept of angle and distance, finite geometry that omits continuity, etc.
Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, a problem that was stated in terms of elementary arithmetic, and remained unsolved for several centuries.
Hey,....
is that correct to say that "The null part added to itself will always remain the same as itself, namely null.""
In arithmetic, 3 * 0 = 0 + 0 + 0 = 0;
It is therefore healthy to extrapolate as follows:
∞-1 * 0 = 0 + 0 + 0 +.... + 0 + 0 + 0 + 0 = 0
∞ * 0 = 0 + 0 + 0 +.... + 0 +...
Dear all,
the following problem is not a home-work problem. I have come up with this question for myself. Nevertheless, I am stuck and need your help.
The question is: Can I calculate the distance between points A and B from this information? And if yes, how?
I think it should be possible...
Consider the following example:
Point A has coordinates 45 lat, 0 long. Point B has coordinates 45 lat, 2 long. Both points are 5000 ft above sea level. The distance between them is X.
Point C has coordinates 45 lat, 100 long. Point D has coordinates 45 lat, 102 long. Both points are at sea...
I want to use this to design a parabolic (optical) mirror;
The problem is that in my application I need both D and f to be a parameter, but I need to specify f only as a perpendicular distance from D. In other words, I need to specify some f_2=f-d, and calculate d. I can't seem to come up with...
Trying to calculate a circumference of a sphere from a radius of 3.09 inches. Is 19.4 a correct answer? Just ran numbers in the first circumference calculator I found http://calcurator.org/circumference-calculator/. Can I use the same formula for a sphere? What can I say ...Geometry is not my...
I have calculated the height of the segment using the Pythagorean Theorem and that's currently where I am right now. I don't seem to find any equations that can help me. Though I might be not trying hard enough or using the wrong words because I'm not really fluent in mathematical terms as you...
This is jut an example to illustrate my doubt. I dont know how to obtain the tracjectory given only the acceleration in this format. I realized that if i can show that there is an constat vector 'a' that satisfy a•r=constant, than the motion would be on the surface of a cone. So i tried to make...
Pictured below are two hinged panels that can rotate upward to form an upside-down V. In position 1, the panels are lying flat. In position 2, the panels have folded together and the joined edge is raised up.
Normally, in order to actuate this hinging motion, one would need to manually lift the...
Hello folks,
I want to simulate a 2D heat transfer process in the subsurface on a region which is infinite on the r-direction. So, as you know, the very basic way to model this is to draw a geometry that is very long in the r direction. I have done this, and the results that I obtain is correct...
I have been working on a problem for a while and my progress has slowed enough I figured I'd try reaching out for some more experience. I am trying to map a point on an ellipsoid to its corresponding point on a sphere of arbitrary size centered at the origin. I would like to be able to shift any...
I'm imagining something like this:
The image was taken from the following paper, and is described as a rhombicuboctahedral quasicrystal. The paper itself gets very technical (at least for me), describing projecting a 4D crystal into 3D space. It seems to me based off of a rhombicuboctahedron...
Here is my attempt to draw a diagram for this problem:
I'm confused about the "the perpendicular bisector of ##BC## cuts ##BA##, ##CA## produced at ##P, \ Q##" part of the problem.
How does perpendicular bisector of ##BC## cut the side ##CA##?
Hello,
I am studying geometry with an app on my phone. There was a difficult problem, which had two different explanations for solving. I correctly understood one explanation. I reviewed later without memory of the problem at all. There was an obvious attempt from what was learned previously...
In ##\mathbb{R}^2##, there are two lines passing through the origin that are perpendicular to each other. The orientation of one of the lines with respect to ##x##-axis is ##\psi \in [0, \pi]##, where ##\psi## is uniformly distributed in ##[0, \pi]##. Also, there are two vectors in...
I want to find the probability that the two points ($x_1, y_1$) and ($x_2, y_2$) lie on the opposite sides of a line passing through the origin $o = (0, 0)$ and makes an angle $\psi$ that is uniformly distributed in $ [0, \pi]$ with the $x$ axis when the angle is measured in clockwise direction...
Is the intersection of a 4D line segment and a 3D polyhedron in 4D a point in 4D, if they at all intersect? Intuitively, it looks like so. But I am not sure about it and how to prove it.
Hi,
I take a big number of disks to composed a circle of a radius of 1 m, the blue curved line is in fact several very small disks:
I take a big number of disks to simplify the calculations, and I take the size of the disks very small in comparison of the radius of the circle. The center A1...
A recipient (cube) of 1m³ is filled of small spheres, there are for example 1000³ spheres inside the recipient. There are also 1000³ elastics that attract the spheres to the bottom. The elastic are always vertical. One elastic for each sphere. One end of the elastic is fixed on a sphere and the...
I am unsure how to go about this. I tried following the suggestion blindly and end up with with some cumbersome terms that are not the answer. From what I understand the derivative at each point would equal to T?
Answer: I just can seem to get to this. I think I'm there but can't get it in...
Consider a point A outside of a line α. Α and α define a plane.Let us suppose that more than one lines parallels to α are passing through A. Then these lines are also parallels to each other; wrong because they all have common point A.
CONTEXT: We are finding the the buoyancy force on a boat which is upright in a still water (Fluid at rest) and the only gravity is acting as the external force. So, first we go for imaging a proper geometry of our boat.
See this figure :
For this figure the book writes:
Fig 8 represents...
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...
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...
I am uncertain if this belongs in the differential geometry thread because I don't know what area of mathematics my question belongs in to 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...
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}...
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.
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 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 equation...
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...
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.
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 +...
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...
Greetings everybody. This is my first post and I am looking for help with a little math/geometry/engineering problem. This has been a real brain buster for my colleague and I the past couple days so I am hoping somebody can help. I am not sure if this is the best section for it, but it...
Homework Statement
I need to obtain the equation of 2 paraboloids separated by a distance L.
Homework Equations
I think that the equations should be:
z_1=x^2+y^2
z_2=x^2+y^2-L
The Attempt at a Solution
The problem is that when I plot the region between two inequations,
x^2+y^2>=z and...
<Moderator's note: Moved from another forum and thus no template.>
I have a project from my Mechanics class which consist on building the system on the picture. It has a spring, cable and a protractor. I have to do an approximation of the mass and weight of the object that is put in the...
I have computed a line from point A to point B by just subtracting the coordinates as below:
line = np.array (x2-x1, y2-y1)
I am not sure if I need to form the formula for this line first by computing the slope and intercept but I continued my code as below, slope being m and b being the...
Hello.
I am wondering how I can find the area of a trapezoid from its two legs and bases.
My problem:
ABCD is a trapezium with AB parallel to CD such that AB = 5, BC = 3, CD = 10 and AD = 4. What is the area of ABCD?
If we trace a straight line from A down parallel to the height of the...
Hi! I'm asked to find all the non-zero Christoffel symbols given the following line element:
ds^2=2x^2dx^2+y^4dy^2+2xy^2dxdy
The result I have obtained is that the only non-zero component of the Christoffel symbols is:
\Gamma^x_{xx}=\frac{1}{x}
Is this correct?
MY PROCEDURE HAS BEEN:
the...
Dear Physics Forums members,
I have a research problem that involves electrostatics. My education is as a chemist, and thus I struggle to accurately represent my problem, so I thought that you guys could help me (and would be interested in the exercise).
Here is an image to summarize my...