We want to find the first intersection point between an arbitrary…

Question Answered step-by-step We want to find the first intersection point between an arbitrary… We want to find the first intersection point between an arbitrary ray and a sphere of arbitrary radius at an arbitrary position in space. (i) List and define all of the parameters required to specify the geometry of the ray and the sphere. [2 marks] (ii) Give an algorithm which returns the desired intersection point (if it exists) and the appropriate normal vector at the intersection point. [5 marks] (b) Describe a method which converts an arbitrary sphere to a triangle mesh at a desired resolution. The desired resolution is specified as a desired number of triangles, D. Your method should produce a number of triangles, N, which is within an order of magnitude 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 properties on N(t) that define a (homogeneous) Poisson process of rate ?. [4 marks] (b) By dividing the interval [0, t] into equal length sub-intervals show that N(t) is a Poisson random variable with mean ?t. [4 marks] (c) Let X1 denote the time of the first event and for n > 1 let Xn denote the elapsed time between the (n ? 1)th and the nth events of the Poisson process. 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 probability density function of the random variable Sn(t). [4 marks] (e) Give an algorithm to generate the first T time units of a (homogeneous) Poisson process of rate ?. [4 marks]  6 Specification and Verification I (a) Explain the difference between a variant and an invariant. Briefly describe what they are used for. [4 marks] (b) State and justify the verification conditions for the total correctness of WHILE commands. [6 marks] (c) (i) Devise a precondition P that makes the following specification true. [P] WHILE I?N DO SUM := SUM+(2×I); 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 logic (LTL) and computation tree logic (CTL). [2+2 marks] Illustrate your answer by contrasting the meanings of G P in LTL with AG P in CTL (where P is a property of states). [2+2 marks] (b) Give an LTL property that cannot be expressed in CTL. [2 marks] (c) Give a CTL property that cannot be expressed in LTL. [2 marks] (d) Describe briefly the kinds of properties that can be expressed using Sugar Extended Regular Expressions (SEREs), Foundation Language (FL) formulae and Optional Branching Extension (OBE) formulae of the Sugar 2.0 property language. [4 marks] (e) Consider the property: “whenever a, b and c occur on successive cycles, then on the cycle that c occurs, d must occur also, followed on the next cycle by e” (where a, b, c, d and e are boolean RKImage transcription textQuestion 2: Stats projects. In a largeIntroductory Statistics lecture hall, theprofessor reports that 5… Show more… Show moreImage transcription text6) The following data separate out the mathematics students at LPC onthe basis of their classes and gender. Algebra Statistics TrigonometryCalculus Total Men 330 115 78 72 Women 345 210 85 65 To… Show more… Show more  Computer Science Engineering & Technology C++ Programming BIOL 174 Share QuestionEmailCopy link Comments (0)