What is the primary purpose of interpolation in computer graphics?

Interpolation is used to estimate points between two given points, allowing for the creation of smooth curves or surfaces.

Fill in the blank: Linear interpolation results in a ________ line connecting two consecutive points.

Polygonal

What mathematical property does linear interpolation preserve when applied to collinear points?

Linear interpolation preserves barycentric coordinates, maintaining the geometric ratio of the points.

True or False: Extrapolation is used to estimate values within a given range of data.

False. Extrapolation is used to estimate values outside the range of given data.

How does the Binomial Theorem facilitate the expansion of binomial expressions?

The Binomial Theorem simplifies expansions.

• Uses coefficients from Pascal's Triangle.

• Efficiently computes large powers.

• Applies to binomial expressions (a+b).

Fill in the blank: The process of estimating values between known data points is known as ________.

Interpolation

Explain what Lagrange's interpolation formula accomplishes.

Lagrange's interpolation formula constructs a polynomial that passes through a given set of unequal data points, providing an exact fit for the data.

What role do forward and backward differences play in Newton's interpolation methods?

Forward and backward differences derive interpolation.

• Estimate function values accurately.

• Use equally spaced data points.

• Handle unequally spaced data effectively.

True or False: The Newton's backward interpolation formula is used when estimating values closer to the start of the interval.

False. It is used for values closer to the end of the interval.

What is meant by 'affine invariance' in the context of piecewise linear interpolation?

Affine invariance maintains interpolant structure.

• Invariant under affine transformations.

• Preserves data relationships intact.

• Essential for accurate interpolation.

Describe the significance of the Menelaus Theorem in geometric interpolation.

Menelaus Theorem is significant.

• Defines collinearity of points.

• Establishes relationships in geometry.

• Crucial for interpolation methods.