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:
To find 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, we will follow these steps:
Step 1: Define the coordinates and values of the end points.
A (0, 5)
B (10, 20)
Step 2: Determine the slope at A.
The slope at A is given as 2.
Step 3: Determine the curvature at A.
The curvature at A is given as -0.5.
Step 4: Define the cubic spline equation as:
S(x) = a + b(x - xi) + c(x - xi)^2 + d(x - xi)^3
Step 5: Find the coefficients a, b, c, and d.
To find the coefficients, we need to solve a system of equations using the conditions at the endpoints.
Condition 1:
S(0) = A(0, 5) = a + b(0 - 0) + c(0 - 0)^2 + d(0 - 0)^3
S(0) = a
Condition 2:
S(10) = B(10, 20) = a + b(10 - 0) + c(10 - 0)^2 + d(10 - 0)^3
S(10) = a + 10b + 100c + 1000d
Condition 3:
S'(0) = 2
S'(0) = b
Condition 4:
S''(0) = -0.5
S''(0) = 2c
Step 6: Solve the system of equations.
Applying Condition 1:
5 = a
Applying Condition 2:
20 = a + 10b + 100c + 1000d
Applying Condition 3:
2 = b
Applying Condition 4:
-0.5 = 2c
Simplifying the equations further, we get:
a = 5
b = 2
c = -0.25
d = 0
Step 7: Substitute the coefficients into the cubic spline equation.
The equation for the cubic spline segment passing through A (0, 5) and B (10, 20), with a slope of 2 and a curvature of -0.5 at A is:
S(x) = 5 + 2(x - 0) - 0.25(x - 0)^2 + 0(x - 0)^3
Simplifying further, we get:
S(x) = 5 + 2x - 0.25x^2