Which of the following statement(s) is/are correct with reference to curve generation ? I. Hermite curves are generated using the concepts of interpolation. II. Bezier curves are generated using the concepts of approximation. III. The Bezier curve lies entirely within the convex hull of its control points. IV. The degree of Bezier curve does not depend on the number of control points.
I, II and III only
Spline curves are either interpolating (the curve passes through every given point) or approximating (the control points only pull the curve towards them).
Statement I — A Hermite curve is an interpolating cubic: it passes through both end points and matches the given tangent vectors there. True.
Statement II — A Bezier curve passes through only the first and last control points; the inner control points shape the curve without lying on it, so it is an approximating curve. True.
Statement III — The Bernstein blending functions are non-negative and sum to 1, so every curve point is a convex combination of the control points; the curve lies inside their convex hull. True.
Statement IV — With \(n + 1\) control points the Bezier curve \(P(u) = \sum_{k=0}^{n} p_k\,BEZ_{k,n}(u)\) has degree \(n\), so the degree is fixed by the number of control points. False.
So I, II and III are correct and IV is wrong. Any choice that includes IV, or leaves out I or III, does not match.
I, II and III only option is correct.
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