(i) List and define all the parameters required to specify the geometry of the ray and the field. [2 marks] (ii) Give an set of rules which returns the favored intersection point (if it exists) and the precise regular vector on the intersection factor. [5 marks] (b) Describe a method which converts an arbitrary sphere to a triangle mesh at a preferred resolution. The favored decision is designated as a preferred wide variety of triangles, D. Your technique ought to produce some of triangles, N, which is inside an order of value of D: D/10 < N < 10D. [4 marks] (c) The Catmull-Clark bivariate subdivision scheme is a bivariate generalisation of the univariate 1 8 [1, 4, 6, 4, 1] subdivision scheme. It creates new vertices as blends of old vertices in the following ways: 16 64 16 64 16 64 16 64 4 64 24 64 4 64 4 64 24 64 4 64 6 64 36 64 6 64 1 64 6 64 1 64 1 64 6 64 1 64 Face Edge Vertex (i) Provide similar diagrams for the bivariate generalisation of the univariate four-point interpolating subdivision scheme 1 16 [1, 0, 9, 16, 9, 0, 1]. [5 marks] (ii) Explain what problems arise around extraordinary vertices (vertices of valency other than four) for this bivariate interpolating scheme and suggest a possible way of handling the creation of new edge vertices when the old vertex at one end of the edge has a valency other than four. [4 marks] 3 [TURN OVER CST.2003.7.4 5 Computer Systems Modelling Let N(t) denote the number of events in the time interval [0, t] for a (homogeneous) Poisson process of rate , ( > 0). (a) State the necessary houses on N(t) that outline a (homogeneous) Poisson system of rate . [4 marks] (b) By dividing the interval [0, t] into equal length sub-periods show that N(t) is a Poisson random variable with imply t. [4 marks] (c) Let X1 denote the time of the primary occasion and for n > 1 permit Xn denote the elapsed time between the (n 1)th and the nth occasions of the Poisson procedure. Determine the distribution of X1 and the joint distribution of X1 and X2. [4 marks] (d) Let Sn = Pn i=1 Xi denote the time of the nth event. Derive the opportunity density characteristic of the random variable Sn(t). [4 marks] (e) Give an algorithm to generate the primary T time units of a (homogeneous) Poisson process of rate . [4 marks] 6 Specification and Verification I (a) Explain the difference among a variant and an invariant. Briefly describe what they may be used for. [4 marks] (b) State and justify the verification situations for the whole correctness of WHILE instructions. [6 marks] (c) (i) Devise a precondition P that makes the subsequent specification true. [P] WHILE IN DO SUM := SUM+(2I); I := I+1 [SUM = N(N+1)] [2 marks] (ii) Devise and justify annotations for this specification that yield provable verification conditions. [8 marks] 4 CST.2003.7.5 7 Specification and Verification II (a) Describe the semantics of formulae in linear temporal good judgment (LTL) and computation tree common sense (CTL). [2+2 marks] Illustrate your answer via contrasting the meanings of G P in LTL with AG P in CTL (where P is a assets of states). [2+2 marks] (b) Give an LTL belongings that can't be expressed in CTL. [2 marks] (c) Give a CTL property that can not be expressed in LTL. [2 marks] (d) Describe briefly the varieties of houses that can be expressed the use of Sugar Extended Regular Expressions (SEREs), Foundation Language (FL) formulae and Optional Branching Extension (OBE) formulae of the Sugar 2.Zero property language. [4 marks] (e) Consider the property: "each time a, b and c arise on successive cycles, then at the cycle that c happens, d must occur also, accompanied on the subsequent cycle by using e" (where a, b, c, d and e are boolean expressions). Use this assets to demonstrate how SEREs can sometimes assist specify residences more compactly than pure LTL. [4 marks] 5 [TURN OVER CST.2003.7.6 eight Information Theory and Coding (a) Describe the types of facts that are amenable to lossy compression with an explanation of why they're amenable to lossy compression. Describe the mechanisms which can be used to perform lossy compression with a proof of why they improve the compression rate. [8 marks] (b) Two coding schemes are proposed for the binary coding of a 4-symbol alphabet: Symbol Code 1 Code 2 a 00 0 b 01 10 c 10 one hundred ten d 11 111 Under what possibility distributions would Code 2 be a greater efficient code than Code 1? You may anticipate that p(a) p(b) p.C) p(d). [6 marks] (c) For an alphabet inclusive of m equiprobable symbols encoded the use of a binary prefix code, show that the common length in keeping with image of the binary code is more than or identical to log2 (m) bits, for any feasible prefix code.One-dimensional Walsh-Hadamard basis features. [4 marks]

Calculate the coefficients of your selected 8 basis functions for the subsequent.RK(i) List and define all the parameters required to specify the geometry of the ray and the field. [2 marks] (ii) Give an set of rules which returns the favored intersection point (if it exists) and the precise regular vector on the intersection factor. [5 marks] (b) Describe a method which converts an arbitrary sphere to a triangle mesh at a preferred resolution. The favored decision is designated as a preferred wide variety of triangles, D. Your technique ought to produce some of triangles, N, which is inside an order of value of D: D/10 < N < 10D. [4 marks] (c) The Catmull-Clark bivariate subdivision scheme is a bivariate generalisation of the univariate 1 8 [1, 4, 6, 4, 1] subdivision scheme. It creates new vertices as blends of old vertices in the following ways: 16 64 16 64 16 64 16 64 4 64 24 64 4 64 4 64 24 64 4 64 6 64 36 64 6 64 1 64 6 64 1 64 1 64 6 64 1 64 Face Edge Vertex (i) Provide similar diagrams for the bivariate generalisation of the univariate four-point interpolating subdivision scheme 1 16 [1, 0, 9, 16, 9, 0, 1]. [5 marks] (ii) Explain what problems arise around extraordinary vertices (vertices of valency other than four) for this bivariate interpolating scheme and suggest a possible way of handling the creation of new edge vertices when the old vertex at one end of the edge has a valency other than four. [4 marks] 3 [TURN OVER CST.2003.7.4 5 Computer Systems Modelling Let N(t) denote the number of events in the time interval [0, t] for a (homogeneous) Poisson process of rate , ( > 0). (a) State the necessary houses on N(t) that outline a (homogeneous) Poisson system of rate . [4 marks] (b) By dividing the interval [0, t] into equal length sub-periods show that N(t) is a Poisson random variable with imply t. [4 marks] (c) Let X1 denote the time of the primary occasion and for n > 1 permit Xn denote the elapsed time between the (n 1)th and the nth occasions of the Poisson procedure. Determine the distribution of X1 and the joint distribution of X1 and X2. [4 marks] (d) Let Sn = Pn i=1 Xi denote the time of the nth event. Derive the opportunity density characteristic of the random variable Sn(t). [4 marks] (e) Give an algorithm to generate the primary T time units of a (homogeneous) Poisson process of rate . [4 marks] 6 Specification and Verification I (a) Explain the difference among a variant and an invariant. Briefly describe what they may be used for. [4 marks] (b) State and justify the verification situations for the whole correctness of WHILE instructions. [6 marks] (c) (i) Devise a precondition P that makes the subsequent specification true. [P] WHILE IN DO SUM := SUM+(2I); I := I+1 [SUM = N(N+1)] [2 marks] (ii) Devise and justify annotations for this specification that yield provable verification conditions. [8 marks] 4 CST.2003.7.5 7 Specification and Verification II (a) Describe the semantics of formulae in linear temporal good judgment (LTL) and computation tree common sense (CTL). [2+2 marks] Illustrate your answer via contrasting the meanings of G P in LTL with AG P in CTL (where P is a assets of states). [2+2 marks] (b) Give an LTL belongings that can't be expressed in CTL. [2 marks] (c) Give a CTL property that can not be expressed in LTL. [2 marks] (d) Describe briefly the varieties of houses that can be expressed the use of Sugar Extended Regular Expressions (SEREs), Foundation Language (FL) formulae and Optional Branching Extension (OBE) formulae of the Sugar 2.Zero property language. [4 marks] (e) Consider the property: "each time a, b and c arise on successive cycles, then at the cycle that c happens, d must occur also, accompanied on the subsequent cycle by using e" (where a, b, c, d and e are boolean expressions). Use this assets to demonstrate how SEREs can sometimes assist specify residences more compactly than pure LTL. [4 marks] 5 [TURN OVER CST.2003.7.6 eight Information Theory and Coding (a) Describe the types of facts that are amenable to lossy compression with an explanation of why they're amenable to lossy compression. Describe the mechanisms which can be used to perform lossy compression with a proof of why they improve the compression rate. [8 marks] (b) Two coding schemes are proposed for the binary coding of a 4-symbol alphabet: Symbol Code 1 Code 2 a 00 0 b 01 10 c 10 one hundred ten d 11 111 Under what possibility distributions would Code 2 be a greater efficient code than Code 1? You may anticipate that p(a) p(b) p.C) p(d). [6 marks] (c) For an alphabet inclusive of m equiprobable symbols encoded the use of a binary prefix code, show that the common length in keeping with image of the binary code is more than or identical to log2 (m) bits, for any feasible prefix code.One-dimensional Walsh-Hadamard basis features. [4 marks]

Calculate the coefficients of your selected 8 basis functions for the subsequent.RK