What is 3d geometry: Definition and 28 Discussions

In mathematics, solid geometry is the traditional name for the geometry of three-dimensional Euclidean space (i.e., 3D geometry).
Stereometry deals with the measurements of volumes of various solid figures (or 3D figures), including pyramids, prisms and other polyhedrons; cylinders; cones; truncated cones; and balls bounded by spheres.

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1. Equal intercept on the three axes

The solution in my book is as follows: "Since the line has equal intercepts on axes, it is equally inclined to axes. \implies line is along the vector a(\hat i + \hat j + \hat k) \implies Equation of line is \frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{1}" As per my understanding, an intercept is the...
2. A,B,C are 3 points with position vectors a,b,c Find length of median

Heres how I proceeded, Equation of line ##AC## in vector form: $$\vec r=a+t(c-a)$$$$\vec r=(1i+4j+3k)+t(2i-6j+2k)$$ Since ##B## doesn't lie on ##AC## ##b\neq (1+2t)i +(4-6t)j+(3+2t)k## The following equation is derived: 2\hat i+\alpha \hat j+4\hat k\neq (1+2t)\hat i +(4-6t)\hat j+(3+2t)\hat k...
3. I Converse of focus-directrix property of conic sections

In my recent study of Conic Sections, I have come across several proofs (many of those comprise Dandelin spheres) showing that the cross-section of a cone indeed follows the focus-directrix property: "For a section of a cone, the distance from a fixed point (the focus) is proportional to the...
4. I Is the Belt Trick Possible with Continuous Deformation in 3D Rotation Space?

Hi, in the following video at 15:15 the twist of ##4\pi## along the ##x## red axis is "untwisted" through a continuous deformation of the path on the sphere 3D rotations space. My concern is the following: keeping fixed the orientation in space of the start and the end of the belt, it seems...
5. I I need help generating a 2D representation of a curved plane

When I was a freshman in college I never thought I'd use geometry... BUT I WAS WRONG! I purchased a laser engraving/cutting machine and my designs keep getting more complex. I have come to the end of what I can figure out or find on the web. I am making a jewelry box for my wife that has a...
6. Tetrahedron with 3 points fixed, and force applied to 4th

My approach to this problem is to recognize that the tetrahedron being still means that net torque is zero and net force is zero. Fd is given Fa + Fb + Fc = -Fd Fa X a + Fb X b + Fc X c = <0,0,0> This can be split up into a series of 6 equations, 2 for each component. However, this is where I...
7. I General Method for Mapping an Ellipsoid to Unit Sphere

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...
8. 3D geometry exercises with linear algebra

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...
9. Efficient solution to a 3D geometry distance problem

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...
10. I Doubling a Cube: Can 3D Geometry Help?

In plane geometry it is impossible to construct a line equal to the (cube root of 2) times the length of a side of a cube, making it impossible to double a cube with a compass and straight edge. Maybe plane geometry needs one more dimension. What happens if we extend the geometry to 3D(solid...
11. A crystallography problem using 3D geometry

Homework Statement Three close-packed planes of atoms are stacked to form fcc lattice. The stacking sequence of the three planes can be altered to form the hexagonal close packed structure by sliding the third plane by the vector r over the second. If the planes in the fcc structure are all...
12. Crystallography; from hcp to fcc

Homework Statement Three close-packed planes of atoms are stacked to form fcc lattice. The stacking sequence of the three planes can be altered to form the hexagonal close packed structure by sliding the third plane by the vector r over the second. If the planes in the fcc structure are all...
13. I Prove that solid angle of any closed surface is 4pi

I googled a lot on proof of Gauss theorem and nearly every other proof (on web and so on books) state that solid angle of closed surface is 4pi but I can't find the proof of this nowhere ! I tried setting up the integral but don't know how to proceed furthur : Ω=∫(cosθ/r^2)*dA Also The one...
14. I Visualizing Solid Angle of a 3d Object (say a Sphere)

Hello Everybody! Concept of Solid Angle was pretty much straight forward until they were on surface patches were taken into account which were visualized as base of cone. I am having difficult when 3d Objects like Sphere/Cylinder . We can very easily calculate the respective area and plugin the...
15. I Spherical Angle Math: Pyramid Angles Constant?

Hi, Calculating the angles in 3D shapes can be a very frustrating and annoying thing. So, I was wondering, are there any mathematical terms, which describe a 3d angles?( "angles" between three lines- part of a sphere) If there are such terms, suppose a triangular pyramid. Is the sum of those...
16. 3D geometry parallelepiped problem

Homework Statement [/B] Given a rectangular parallelepiped ABCDEFGH, the diagonal [AG] crosses planes BDE and CFH in K and L. Show K and L are BDE's and CFH's centres of gravity. I think I have understood the problem, could you verify my demo please ? Thanks Homework Equations The Attempt at...
17. Solving 3D Geometry Problems with Python

Homework Statement Requirements: http://i.imgur.com/2WKyhto.png Homework Equations 2(L * W + L * H + W * H) SA = (2 * pi * radius * height + 2 * pi * radius^2) height = (volume)/(pi * radius^2) The Attempt at a Solution Code link: http://pastebin.com/sKFEGN0C
18. Is a line observable(in real life)?

can a line (thickness only in 1 dimension ) be observed in real life ? since ,at micro level , even atoms have thickness, is it even possible to really construct a line or point or plane in reality?
19. 3D object represent with primitive shapes

Hi, Given a 3D object in R3 space can we represent it using three basic primitive shapes like Sphere, Cone and Cylinder? Would this claim be valid?
20. 3D geometry: parametric equation and tangents

I have a doubt in 3d geometry. I calculus and I know how to do partial derivatives.(but I don't know what it means). If you have a parametric equation ##x=t, y=t^2,z=t^3## (the equation is randomn) What does ##\vec{r}=t\hat{i}+t^2\hat{j}+t^3\hat{k}## represent? now if it represents the position...
21. ± sqrt(b^2 - a^2) = ± b ± ai ?

I came across this strange relationship when deriving the degree-4 equation for a torus. First thing that comes to mind is the 'Freshman's Dream'. Apparently, it was pure coincidence that they are equal. But, I don't believe in coincidences when it comes to a math expression. There is something...
22. Software for 3D geometry illustrations?

Dear all, I am looking for an interactive software that can let me play with 3d geometry. I apologize, as I am not sure of the right technical term of the sorts of shapes I am interested to work with. I am not a mathematician or physicist, but rather use these as philosophical metaphors and...
23. 3D Geometry: Calculate the position of the north star

Homework Statement Ok I'm able to track one star in the sky, over a period of one hour. We use three measurements to find the angle of the arc traced by the star. The three measurements also constitute two vectors. We can take the cross product of those vectors and it will give us a vector...
24. Discover the Length of a Sidereal Day with Just a Star and Time Measurements

Homework Statement a. Suppose you do not know your location (on Earth) or the direction of north. Now suppose you track one particular star in the sky. You measure its exact position in the sky and record the exact time of the measurement. How many such measurements are necessary to deduce...
25. When two spheres intervine - very hard 3d geometry and vectors problem

Homework Statement Sphere with P as centre has equation (x-5)^2 + (y-9)^2 + z^2 -100 = 0 Sphere with Q as centre has equation (x-1)^2 + (y+3)^2 + (z-3)^2 -49 =0 These spheres intervine with each other. Find the surface area of the object limited by the two spheres. The Attempt at a Solution...
26. What is the rate of change of angle for a knight jumping on a chess board?

Some of my friends, upon finding out that I'd joined these forums, decided to give me a math challenge problem they thought couldn't be solved. Does everyone get the same answer? It goes as follows: A knight jumps on a chess board (assume it makes a legal move) from point A to point B...
27. Help needed with 3d geometry problem: important

help needed with 3d geometry problem: important! Homework Statement find the equations of the two planes through the origin which are parallel to the line (x - 1)/2 = (-y-3) = (-z-1)/2 and at a distance of 5/3 from it. also, show that the planes are perpendicular Homework Equations...
28. Visualizing 3D Geometry Through Algebraic Solutions

So each of three formulas has three variables. I am trying to picture this geometrically as if it were position variables in 3D space. Then each formula describes a unique point in 3D space, and the row vector goes from the 000 point to the point described by the formula. So, there are three...