Q. Determine the equation for a cubic spline segment passing through two end points A (0, 5) and B (10, 20), having a slope of 2 and a curvature of -0.5 at A. Equation for cubic spline is: y=a0+a1.x+a2.x^2+a3.x^3 Slope equation is:y’=dy/dx=a1+ 2a2.x+3a3.x^2 curvature equation is:y”=d2y/dx2=2a2+6a3.x
To determine the equation for a cubic spline segment passing through two end points A(0, 5) and B(10, 20), with a slope of 2 and a curvature of -0.5 at A, we can use the following steps:
Step 1: Determine the values of a0, a1, a2, and a3.
Since we have the equation y = a0 + a1.x + a2.x^2 + a3.x^3 for the cubic spline segment, we need to find the values of a0, a1, a2, and a3.
Step 2: Use the given conditions to form equations.
We are given the following conditions:
- The segment passes through points A(0, 5) and B(10, 20).
- The slope at A is 2.
- The curvature at A is -0.5.
Using the equation y = a0 + a1.x + a2.x^2 + a3.x^3, we can substitute the x and y values for A and B to form two equations:
For point A: 5 = a0 + a1(0) + a2(0)^2 + a3(0)^3
=> 5 = a0
For point B: 20 = a0 + a1(10) + a2(10)^2 + a3(10)^3
=> 20 = a0 + 10a1 + 100a2 + 1000a3
Step 3: Use the slope equation to form another equation.
We are given that the slope at point A is 2. The slope equation, which is the derivative of the equation for the cubic spline segment, is y' = a1 + 2a2.x + 3a3.x^2.
Substituting x = 0 and y' = 2 for point A, we get:
2 = a1 + 2a2(0) + 3a3(0)^2
2 = a1
Step 4: Use the curvature equation to form another equation.
We are given that the curvature at point A is -0.5. The curvature equation, which is the second derivative of the equation for the cubic spline segment, is y'' = 2a2 + 6a3.x.
Substituting x = 0 and y'' = -0.5 for point A, we get:
-0.5 = 2a2 + 6a3(0)
-0.5 = 2a2
Step 5: Solve the system of equations.
Using the equations obtained from steps 2, 3, and 4, we can solve for the values of a0, a1, a2, and a3.
From step 2, we know that a0 = 5.
From step 3, we know that a1 = 2.
From step 4, we know that a2 = -0.25.
Step 6: Substituting the values of a0, a1, a2, and a3 into the equation.
Using the values obtained from step 5, we can substitute a0 = 5, a1 = 2, a2 = -0.25, and a3 = 0 into the equation y = a0 + a1.x + a2.x^2 + a3.x^3.
The equation for the cubic spline segment passing through the given points, with the given slope and curvature at point A, is:
y = 5 + 2x - 0.25x^2
Therefore, the equation for the cubic spline segment is y = 5 + 2x - 0.25x^2.