How can I use Bezier curves to plot points in relative space?

[PerlTrader]

Hello math gurus,

I'm using Bezier curves in my Perl application to chart currency trading data.
Having no formal math training, I've framed my questions in a modeling context.
I want to step along the major axis (on integer boundaries), and calculate real
values on the minor axis. I'm plotting in relative space, like PostScript's
"rcurveto" function. The example code fragments are in (pseudo) Perl with
scalar($) and termination(;) characters removed.

Control Points:
x1 = 70 y1 = 0
x2 = 100 y2 = 20
3 = 100 y3 = 50

Initialize Curve:
cX = (3 * x1)
cY = (3 * y1)
bX = ((3 * (x2 - x1)) - cX)
bY = ((3 * (y2 - y1)) - cY)
aX = (x3 - cX - bX)
aY = (y3 - cY - bY)

Plot Point:
t1 = t t2 = (t1 * t1) t3 = (t1 * t2)
pX = ((aX * $t3) + (bX * $t2) + ($cX * t1))
pY = ((aY * $t3) + (bY * $t2) + ($cY * t1))

Octant Change:
T: 0.65429213786 X: 88.830577627 Y: 22.884879224

Octant 0 Point:
T: 0.28268189551 X: 50.000000000 Y: 4.568654814

Octant 1 Point:
T: 0.88425060334 X: 98.778678931 Y: 40.000000000

In Octant 0, I want to use X integers (0 to 88) to find t (e.g. 0.28268189551
from 50) to generate Y reals (e.g. 4.568654814 from 0.28268189551). In Octant 1,
I want to use Y integers (23 to 50) to find t (e.g. 0.88425060334 from 40) to
generate X reals (e.g. 98.778678931 from 0.88425060334).

My three part question is:

1) Can the (aX, bX, cX, aY, bY, cY) initialization values be used in the solution,
or will separate initialization values be required for X stepping and Y stepping.

2) Are the X and Y stepping solutions (that find t) similar to the structure of
finding X and Y with t?

Find t with x: t1 = pX ... process ...
Find y with t: ((aY * $t3) + (bY * $t2) + ($cY * t1))

3) How will the solution calculate the following values from X = 50 in Octant 0
and Y = 40 in Octant 1:

T: 0.28268189551 X: 50.000000000 Y: 4.568654814

The only way I'll be able to understand a solution is if I can plug it into some
Perl code. Explanations in pseudo code or C fragments should also work.

Thanks (in advance, for anyone willing to help me with this).

 

[jvicens]

Thank you for reaching out with your questions about using Bezier curves in your Perl application. As a scientist with a background in mathematics, I would be happy to help you find solutions for your modeling needs.

Firstly, regarding your question about using the same initialization values for both X and Y stepping, the short answer is yes. The same values for (aX, bX, cX, aY, bY, cY) can be used for both solutions. The key is to understand that the Bezier curve is a parametric curve, meaning that the coordinates (X,Y) are defined in terms of a parameter t. So as long as you are using the same parameter t in both solutions, the initialization values will work for both X and Y stepping.

Secondly, the structure of finding X and Y with t is indeed similar. In both cases, you are using the same formula to calculate the coordinate (pX, pY) from the parameter t. The only difference is that in one case, you are solving for the X coordinate and in the other case, you are solving for the Y coordinate. So as long as you use the same parameter t for both solutions, the structure will be the same.

Finally, to calculate the values T, X, and Y from X = 50 in Octant 0 and Y = 40 in Octant 1, you can follow the same steps outlined in your example code. The only difference is that you will be solving for the parameter t instead of the coordinates (pX, pY). Once you have the value of t, you can then plug it into the formula for pX and pY to get the desired values.

I hope this helps clarify your questions and aids in your modeling process. Please let me know if you have any further questions or if you need any assistance with implementing these solutions in your Perl code.

 

[tacman]

Bezier curves are a powerful tool for creating smooth and precise curves in computer graphics and modeling. They are defined by a set of control points and a mathematical formula that calculates the position of points along the curve based on a parameter, typically denoted as t. In your case, you are using Bezier curves to plot points in relative space, which means that the curve is defined by relative coordinates rather than absolute coordinates. This allows for more flexibility and easier manipulation of the curve.

To use Bezier curves to plot points in relative space, you will need to follow a few steps:

1. Define your control points: In your example, you have three control points (x1, y1), (x2, y2), and (x3, y3). These points will determine the shape of your curve.

2. Initialize the curve: Use the following formula to calculate the coefficients for your Bezier curve:
cX = (3 * x1)
cY = (3 * y1)
bX = ((3 * (x2 - x1)) - cX)
bY = ((3 * (y2 - y1)) - cY)
aX = (x3 - cX - bX)
aY = (y3 - cY - bY)

These coefficients will be used in the next step to calculate the position of points along the curve.

3. Plot points: To plot a point on the curve, you will need to use the following formula:
pX = ((aX * t^3) + (bX * t^2) + (cX * t))
pY = ((aY * t^3) + (bY * t^2) + (cY * t))

Here, t represents the parameter that determines the position of the point on the curve. In your example, you are using t1, t2, and t3 to calculate the position of points. These values are simply t raised to different powers. For example, t2 represents t squared (t^2).

4. Octant change: In your example, you mention octants, which refer to different sections of the coordinate plane. The formula for calculating the position of points in different octants is the same, but the values of t will change. In Octant 0, you will use X integers to find t and generate Y reals, while in Octant 1, you will use