Parametric and Continuity Equations

Question 1

Parametric coordinate functions,Each section of spline is described with parametric coordinate functions,write those functions

Answer 1

x=x(u)
y=y(u)
z=z(u)

u1<=u<=u2

Question 2

What is zero-order Parametric (C0)?

Answer 2

Simply means that the curves meet
x,y and z evaluated at u2 for the first curve section are equal to the values of x,y and z evaluated for the next curve section

Question 3

First-order Parametric C1

Answer 3

The first parametric derivatives( tangent lines) of the coordinate functions for two successive curve sections are equal at their joining point

Question 4

Second-order parametric functions(C2)

Answer 4

Both the first and second parametric derivatives of the two curve sections are the same intersection

Question 5

Three methods for specifying a spline representation.

Answer 5

1. We can state the set of boundary conditions that are
   imposed on the spline.
2. We can state the matrix that characterizes the spline.
3. We can state the set of blending functions

Question 6

We can write the boundary conditions in matrix form and solve for the
coefficient matrix C as:

Answer 6

𝐶 = 𝑀𝑠𝑝𝑙𝑖𝑛𝑒. 𝑀𝑔𝑒𝑜m

Question 7

we can substitute the matrix representation for C into the equation to
obtain

Answer 7

X(u)= 𝑈. 𝑀𝑠𝑝𝑙𝑖𝑛𝑒. 𝑀𝑔𝑒𝑜m

Question 8

The matrix 𝑀𝑠𝑝𝑙𝑖𝑛𝑒, characterizing a spline representation, sometimes
called the

Answer 8

the basis matrix

Question 9

what are gk and BFk in the polynomial representation of the parameters of the geometric constraint

Answer 9

gk control point coordinates and slope of the curve

BFk polynomial blending functions