1. Bending in 2 Planes Parametric form assuming small deflections and no torsion.

2. Parametric Form Where Condensed:

3. Deflection in Y & Z Sensor 1 Sensor 2 x1x1 x2x2 L = 15cm Tip Deflection y x z

4. Cross-section at Sensors y z x Known: At u = 0: At u = 1: Seven unknowns remain.

5. Curvature Information Assuming no torsion, the orientation of the local and global coordinate frames are the same. From each known curvature component, we have an equation with 5 unknowns. For example, at u = 1: Therefore, at each sensor two orthogonal components of curvature are known, and 2 equations with 5 unknowns result. (y”(x)  a1x, a2x, a3x, a2y, a3y and z”(x)  a1x, a2x, a3x, a2z, a3z). Therefore two sets of curvatures are needed.

6. To Simplify Math Assume small deflections in Y and Z, such that Then, We have rid of 3 more unknowns (x coefficients).

7. Simplified Derivatives The parametric slope equations are now: The parametric curvature equations are now:

8. Parametric Equations in Terms of x Substitute in for u:

9. Parametric Derivatives in Terms of x Slope equations: Curvature equations: Now, the equations to solve for the four remaining unknown coefficients are simple. y”(x) and z”(x) are the known orthogonal components of the curvature at the sensor locations (and at the tip).

10. Model Find y’’(x) at sensor locations. Calculate a set of unknowns for the region from the base to x2 (second sensor location.) Estimate y(x2) and y’(x2). Use y(x2), y’(x2) and y’’(L) to calculate a set of unknowns from the region from x2 to the tip of the needle.