MAT Midterm Exam Summer 7 Note: For True-False questions, a statement is only True if it must always be True under the given assumptions, otherwise it is False.. The control points of a Bezier curve γ(t) of degree d are: i) values of γ(t) for t ii) parametric polynomials iii) point-coefficients of γ(t) with respect to the Bernstein basis FTF b) TTT c) TFT d) FFT TTF FFT. A linear combination of polynomials p (t),..., p n (t) is a sum a p (t)+ +a n p n (t), with real number coefficients a i. The derivative of a Bernstein polynomial of degree d can be written as a linear combination of: i) two Bernstein polynomials of degree d ii) Bernstein polynomials of degree d iii) two Bernstein polynomials of degree d FTF b) TTT c) FFT d) FFF TFT FFT. Let P = (, ), P = (, ), P = (, ), P = (, ), and P = (, ), and let γ(t) be the Bezier curve with control points P, P, P, P, and P. i) γ(t) has degree ii) γ() = γ() iii) γ () γ () = TTT b) FFF c) TTF d) FTT FTF TTF. Same γ(t) as in the previous question. Find the Bezier point P which is used to compute γ( ). (Sketch the NLI steps.) (, ) b) (, ) c) (, 7 ) d) (, ) (, 5 ) (, ) 5. Same γ(t) as in the previous question. Find γ( ). (, ) b) (, ) c) (, 7 ) d) (, ) (, 5 ) (, 5 ) 6. Suppose g() =, g() =, and g(5) = 6. Find the divided difference: [,, 5]g: b) 5 c) d) 6 5
7. Recall that a function f has a zero of multiplicity r at t = c if f (k) (c) = for k =,..., r. We say that f has a zero of exact multiplicity r if additionally f (r) (c). Suppose the Bernstein polynomial Bi d (t) has a zero of exact multiplicity 5 at t = and a zero of exact multiplicity at t =. i) d = 5 ii) d = iii) d i = 5 FFT b) FFF c) FTF d) TTT TTF FFF. Let γ(t) = γ [P,P,P ](t) be the quadratic Bezier curve with control points P = (, ) and P = (, ), and all points of γ(t) lying on the parabola y = x. Find P. (, ) b) (, 6) c) (, 5) d) (, ) (, ) (, 5) 9. Compute the following Vandermonde determinant: 5 5 5 5. 6 6 6 6 7 7 7 7 5 b) c) 6 5 d) 7 5 5 6 5. Let Bi d(t) = ( d i) ( t) d i t i be the general Bernstein polynomial, and suppose that v = is the coordinate vector of a polynomial p(t) with respect to the Bernstein basis of P. Find the coordinate vector of p(t) with respect to the shifted basis {, t, (t ) }. b) c) d). Suppose that v = is the coordinate vector of a polynomial p(t) with respect to the standard basis of P. Find the coordinate vector of p(t) with respect to the Bernstein basis of P. (Hint: solve for the two standard basis vectors that sum up to this vector, since they each have simple Bernstein representations.) b) c) d). Let F [u, u, u, u ] be the polar form of the polynomial t in P. Find F [,,, ]. b) c) d)
. Let F [u, u, u, u ] be the polar form of the polynomial t in P. Find F [,,, ]. b) c) d). Let γ(t) = (t t, t + t ). Use the polar form of γ(t) with respect to the standard basis to find the control point P. (, ) b) (, ) c) (, ) d) (, ) (, ) (, )
5. Let g(t) = (t ) (t ), and let G[u, u, u ] = (u )(u )(u ) + (u )(u )(u ) + (u )(u )(u ). For G to be the polar form of g it would need to satisfy the three defining properties of polar forms: i) symmetry, ii) substitution, and iii) affine. Determine if the statements are True or False: i) G satisfies i) ii) G satisfies ii) iii) G satisfies iii) TFT b) TTT c) FTT d) FFF TTF TFT 6. Let q(t) be a polynomial of degree at most such that q() =, q () =, q () = 6, and q() =. What is q()? (Don t forget to divide by when entering the second derivative into the divided difference table.) b) c) d) 9 9 7. If p(t) = + (t ) + 5(t )(t ) is the Newton form of the interpolating polynomial that matches a data function g(t) for data values t =, t =, and t =, what is [, ]g? b) 5 c) d) 6. Same p(t) and g(t) as in the previous question. Suppose that another interpolation point is added so that q(t) is the new interpolating polynomial in P which satisfies the same information as p(t) and also: q() =. Find [,,, ]g: 6 b) c) d) 7 7 9. Find a quadratic Bezier curve γ(t) which agrees with the graph of y = x at the points (, ) and (, ) and also has the same tangent slopes as this graph at those points. If γ() = (, ) and γ() = (, ) find the control point P. (, ) b) (, ) c) (, ) d) (, ) (, 9 ) (, ). Let γ(t) = ( t + t, t + t ). Use the Polar form to find the control points of γ(t). What is P? (, ) b) (, ) c) (, ) d) (, ) (, ) (, ). Let S be the set {(t ), t, t + } of polynomials in P. Determine whether the following statements are True or False. The answers are in order i),ii), iii). i) S is a basis of P ii) S spans P iii) S is linearly independent FFT b) TTT c) FFF d) TFT TFF FFF. Find the derivative: d dt B (t). 6( t)( t) b) t( t) c) t( t) d) t( t) t( t)( t) t( t). Solve for a : (t )(t ) + = a (t ) + + a (t ) +. b) c) d)
. To find an implicit quadratic curve f(x, y) = L (x, y)l (x, y) + cl(x, y) = for a Bezier curve with control points P = (, ), P = (, ), and P = (, ), we could use which of the following implicit forms for f(x, y): x(x + y) + c(x y) b) x(x y) + c(x + y ) c) x(y ) + c(y ) d) (x + y )(y x) + cx xy + c(x y ) (x + y )(y x) + cx 5. Same curve as in the previous question. Solve for c: b) c) d) 9 9