Interpolation

Table of Contents

Why do we need this?

Interpolation can help you construct values between two points based on available data. Common Use Case:

Sensor Data Fusion: Often Sensors operate at different frequency from each other, so the data you might try to collect might not match for a particular timestamp. You can use Interpolation to generate Data for the missing time stamp based on available data from previous records.

LERP: Linear Interpolation

Assume a situation with 2 vectors vi and vf as follows:

Initial-Situation

LERP is used when you want a linear interpolation from start to end. This is the shortest distance between the two points as shown:

Assume you want to find a point on this new vector vfi at some distance from vi which is denoted by Vector vs from the origin.

Here,

\[ s ∈ [0,1]\\ \begin{equation} \begin{split} v_{s} = v_{i} + s(v_{f} - v_{i})\\ v_{s} = (1-s)v_{i} + sv_{f} \end{split} \end{equation} \]

Thus,

\[ \begin{split} \forall s=0 \Rightarrow v_{s} = v_{i}\\ \forall s=1 \Rightarrow v_{s} = v_{1} \end{split} \]

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Time Version for LERP

Lerp(p0,p1;t) = (1−t)p0 + tp1

for t ∈ [0,1]

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This works best in following situations:

The Interpolation is needed in a 3D or less Cartesian System where the interpolation is estimated to be a straight line.

The faster computation is essential over accuracy.

The data points are so close to each other that the ∂X can be approximated to a straight line.

SLERP: Spherical Linear Interpolation

Assume a situation with 2 vectors vi and vf on the surface of the sphere of unit distance from center of the sphere as follows:

SLERP is used when you want an interpolation along an arc between vf and vi. A hypothetical plane (Plane OFI) is constructed to image the motion that passes through the point on sphere for vi, vf and the origin of the sphere. We assume a vector vs along the arc on the same plane OFI.

Assume a slice from the plane as shown:

Here:

• Let the angle between vf and vi be θ.
• If we want to find a vector vs on the resultant arc, the angle of vs w.r.t. vi is sθ.

vs = αsvi + βsvf

If we take a closer look at the defined system, we have the understanding that the vectors are all of equal magnitude with a common origin. i.e. Vector vf and vector vs can be written in terms of a rotation matrix R and vi.

\[ \begin{split} v_{f} = v_{i}R\\ v_{s} = v_{i}R^{s} \end{split}  \]

where: s ∈ [0,1]

Thus,

\[ \begin{split} v_{s} = \alpha_{s}v_{i} + \beta_{s}v_{f}\\ v_{i}R^{s} = \alpha_{s}v_{i} + \beta_{s}v_{i}R \end{split} \]

Upon solving this based on the video attached in resources, the final result seems as follows:

\[ v_{s} = \frac{\sin((1-s)\theta)}{\sin(\theta)}v_{i} + \frac{\sin(s\theta)}{\sin(\theta)}v_{f}  \]

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Time Version for SLERP

\[ Slerp(p_{0}, p_{1};t) = \frac{\sin((1-t)\theta)}{\sin(\theta)}p_{0} + \frac{\sin(t\theta)}{\sin(\theta)}p_{1}  \]

where, θ = cos−1(p0⋅p1)

for t ∈ [0,1]

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Time Version for Quaternion SLERP

Slerp(q0,q1;t) = (q1q0−1)tq0

for t ∈ [0,1]

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This works best in following situations: - The Interpolation is needed in a 3D or less Polar System where the interpolation is estimated to be along a curve. - This process is computationally more intensive than LERP.

Resources

• Geometric Algebra - Linear and Spherical Interpolation (LERP, SLERP, NLERP)
• Wikipedia: Slerp