Locating Points in 3D Space Based on Given Parameters

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In summary: Once you have the distances and angles, you can use basic trigonometry to find the coordinates of the points in 3D space. In summary, the conversation discusses locating points in 3D Cartesian coordinates using given parameters of distance, angles, and torsion angle. The speaker wants to program a simulation to attach atoms using this information and is seeking help with the math behind it. They also mention the use of dot product and basic trigonometry in finding the coordinates.
  • #1
joinforfun89
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Hi,

Let's think about 4 points A,B,C,D.
I need to locate the points in 3D Cartesian coordinates provided the following parameters are given about the point:
1] Distance between A and B.
2] Distance between B and C
3] The angle between A B and C.
4] The angle between B C and D.
5] The torsion angle A-B-C-D.

I want to program this up so that in my simulation I can attach atoms to each other provided the bond-lengths, bond angles and torsion angles are specified. I need help with the math. We know that we must start by setting up the coordinate system. So for the first set of 4 atoms, we can place A at the origin, B along any of the coordinate axes and finally C in a chosen plane. But once we are through with the first set, what is a general formulation for the solution of the problem ?

Please let me know if you know of any books/articles which deal with similar issues.

Thanks for your help.
 
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  • #2
joinforfun89 said:
Hi,

Let's think about 4 points A,B,C,D.
I need to locate the points in 3D Cartesian coordinates provided the following parameters are given about the point:
1] Distance between A and B.
2] Distance between B and C

3] The angle between A B and C.
4] The angle between B C and D.
5] The torsion angle A-B-C-D.
I don't understand what you mean by these. Three points, A, B, and C, don't have a single angle. Assuming you meant the ray, AB, there still is no one angle between a line or ray and a point. Did you mean the angle between rays AB and AC? By "the torsion angle A-B-C-D" I think you mean "the angle around line AB that maps line AC onto AB" but I'm not sure of that.

I want to program this up so that in my simulation I can attach atoms to each other provided the bond-lengths, bond angles and torsion angles are specified. I need help with the math. We know that we must start by setting up the coordinate system. So for the first set of 4 atoms, we can place A at the origin, B along any of the coordinate axes and finally C in a chosen plane. But once we are through with the first set, what is a general formulation for the solution of the problem ?

Please let me know if you know of any books/articles which deal with similar issues.

Thanks for your help.
 
  • #3
HallsofIvy said:
I don't understand what you mean by these. Three points, A, B, and C, don't have a single angle. Assuming you meant the ray, AB, there still is no one angle between a line or ray and a point. Did you mean the angle between rays AB and AC? By "the torsion angle A-B-C-D" I think you mean "the angle around line AB that maps line AC onto AB" but I'm not sure of that.

Yes I mean the ray AB and AC and the smaller angle that is made. The torsion angle can be thought of as the angle around line BC that maps line AB on to CD. In other words, the angle between the planes defined by points A,B,C and the plane defined by points B,C and D.


Thanks.
 
  • #4
Distances between two points A and B can be found by taking the vector A - B and finding its length.

The smallest angle between two vectors can be found using using the dot product. Wiki it. It's a pretty straightforward formula.
 
  • #5


Hi,

Thank you for reaching out. I can provide some guidance on locating points in 3D space based on the parameters you have provided.

To start, we can use the distance formula to determine the coordinates of points A, B, and C. This formula, √(x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2, will give us the distance between any two points in 3D space. We can use this formula to calculate the coordinates of B and C based on the distances given between A and B, and B and C.

Next, we can use the dot product and cross product to determine the angles between points A, B, and C. The dot product, A • B = |A||B|cosθ, will give us the angle between two vectors, where |A| and |B| are the magnitudes of the two vectors and θ is the angle between them. The cross product, A x B = |A||B|sinθ, will give us the vector perpendicular to both A and B, which we can use to determine the angle between B and C.

For the torsion angle, we can use the dot product and cross product again to determine the angle between points A-B-C and B-C-D. This will involve using the vectors from A to B, B to C, and C to D.

As for a general formulation, it would depend on the specific coordinates and angles given for each set of points. However, the distance formula, dot product, and cross product should be able to help you determine the coordinates and angles for any set of points in 3D space.

As for resources, I would recommend looking into textbooks on linear algebra and vector calculus, as they cover the concepts and formulas needed for this type of problem. You may also find helpful resources online, such as tutorials or practice problems.

I hope this helps and good luck with your simulation! Let me know if you have any further questions.
 

What is the Cartesian coordinate system?

The Cartesian coordinate system is a method of locating points in a three-dimensional space using three axes: x, y, and z. The intersection of these axes creates a point of origin, and the coordinates of any point can be determined by measuring its distance from the origin along each axis.

How do you find the distance between two points in 3D space?

To find the distance between two points in 3D space, you can use the Pythagorean theorem. First, determine the differences between the x, y, and z coordinates of the two points. Then, square each difference, add them together, and take the square root of the sum. This will give you the distance between the two points.

What is the difference between absolute and relative position?

Absolute position refers to the exact coordinates of a point in 3D space, measured from the origin. Relative position, on the other hand, refers to the position of a point in relation to another point or object. It is often described using terms such as "above," "below," "to the left of," etc.

How do you convert between Cartesian and spherical coordinates?

To convert from Cartesian coordinates (x, y, z) to spherical coordinates (r, θ, φ), first calculate the distance from the origin using the Pythagorean theorem (r = √(x² + y² + z²)). Then, determine the elevation angle (θ = arccos(z/r)) and the azimuth angle (φ = arctan(y/x)). To convert from spherical coordinates to Cartesian coordinates, use the equations x = r*sin(θ)*cos(φ), y = r*sin(θ)*sin(φ), and z = r*cos(θ).

What is the significance of the origin in 3D space?

The origin (0, 0, 0) serves as the reference point for the Cartesian coordinate system in 3D space. It is the point at which all three axes intersect and is used as the starting point for measuring the position of other points. The origin is also important in many mathematical calculations and equations involving points in 3D space.

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