Construction with ruler and compass:
Plato (427-347 B.C.) considered ruler and compass the merely appropriate tools for geometric construction.
A) Primitive objects: straight line (finite or unbounded), point and circle.
B) Primitive operations:
• Given 2 points P, Q construct a line passing via P and Q• Given 2 points P, Q construct a circle with center P which passes via Q• Given 2 lines L, L’and two circles C, C’ or a line L and a circle C, • Make a new point P at any intersection of such a pair.
C) Compound object: Any structure built from the primitive objects by following a series of primitive operations.
Illustrations:
A) Given a line L and a point P on L, then construct a line L’ passing via P and orthogonal to the L.
B) Construct a pair of Lines L, L’ which defines angles of 90º, 60º, 45º and 30º respectively.
C) The segment is a triple (L, P, Q) with points P and Q on L. The segment of length 1 might be given or it might be randomly constructed to define the unit of measurement.The angles and lengths which can be constructed:
A) Beginning with given angles, construct additional angles through bisection, subtraction and addition.
B) Beginning with a segment of unit length, construct the additional lengths through bisection, addition, subtraction, division, multiplication and square root.
Therefore, construction by ruler and compass is reduced the question whether certain desired quantity can be stated in terms of rational operations and square roots.
Theorem: The proposed construction is probable by ruler and compass if the numbers that define analytically the preferred geometric elements can be derived from such defining the given elements by the finite number of rational operations and extractions of the real square roots.
Possible construction: Equilateral polygon of 17 sides, ‘17-gon’ (that is, Carl Friedrich Gauss 1777-1855)
Impossible constructions: Equilateral heptagon, ‘7-gon’ (Gauss). Doubling the unit cube (simplify x3 = 2).
Theorem: There is no algorithm to trisect the arbitrary angle with the ruler and compass. And yet, the Archimedes (~287 B.C. - ~ 212 B.C) found a process to trisect the arbitrary angle:
Given any angle AOB = 3x in a unit circle. The ruler CB has a mark D at distance CD = radius = r = 1. Slide C all along the x-axis till D lies on circle. Then, angle ACB = x.
Msg: Minor modifications in a model of computation might have drastic consequences- precise definition required!
Example: The quadratic equation x2 + bx + c = 0 consists of roots x1, x2 = (-b ± sqrt(b2 - 4c))/2. Now prove that in the ruler and compass construction illustrated below, the segments x1 and x2 are solutions of this equation.
Solution: Replace xc = - b/2 and yc = (1 + c)/2 in the circle equation (x - xc)2 + (y - yc)2, set y = 0 to get the quadratic equation x2 + bx + c = 0 whose roots are the preferred values x1 and x2 .
Latest technology based Theory of Computation Online Tutoring Assistance
Tutors, at the www.tutorsglobe.com, take pledge to provide full satisfaction and assurance in Theory of Computation help via online tutoring. Students are getting 100% satisfaction by online tutors across the globe. Here you can get homework help for Theory of Computation, project ideas and tutorials. We provide email based Theory of Computation help. You can join us to ask queries 24x7 with live, experienced and qualified online tutors specialized in Theory of Computation. Through Online Tutoring, you would be able to complete your homework or assignments at your home. Tutors at the TutorsGlobe are committed to provide the best quality online tutoring assistance for Theory of Computation Homework help and assignment help services. They use their experience, as they have solved thousands of the Theory of Computation assignments, which may help you to solve your complex issues of Theory of Computation. TutorsGlobe assure for the best quality compliance to your homework. Compromise with quality is not in our dictionary. If we feel that we are not able to provide the homework help as per the deadline or given instruction by the student, we refund the money of the student without any delay.
a counter is a device that stores (and occasionally displays) the number of times a specific event or process has happened, frequently in relationship to a clock signal.
tutorsglobe.com types of natural resources assignment help-homework help by online natural resources tutors
Theory and lecture notes of Extreme Situations-Financial Crises all along with the key concepts of extreme situations: financial crises, Lenders of Last Resort, Deposit Insurance and Moral Hazard. Tutorsglobe offers homework help, assignment help and tutor’s assistance on Extreme Situations-Financial Crises.
leaves of nicotiana tabacum (tobacco) consist of alkaloids nicotine, nornicotine and anabasine.
www.tutorsglobe.com offers Comments on Factory Overheads homework help, assignment help, case study, writing homework help, online tutoring assistance by accounting tutors.
Theory and lecture notes of Profit Maximization all along with the key concepts of Profit maximizing quantity, Deriving Marginal Revenue Curve, Amoroso-Robinson Formula, Marginal Revenue, Price Elasticity, Total Revenue. Tutorsglobe offers homework help, assignment help and tutor’s assistance on Profit Maximization.
Theory and lecture notes on the concept of Frequency Response Considerations all along with the key concepts of Single Inverting Gain phase, Feedback and Stability, Two-Stage Non-Inverting Amplifier, Gain Margin and Phase Margin. Tutorsglobe offers homework help, assignment help and tutor’s assistance on Frequency Response Considerations.
tutorsglobe.com properties of copper sulphate assignment help-homework help by online copper sulphate tutors
tutorsglobe.com bio-war assignment help-homework help by online crop diseases and their control tutors
Theory and lecture notes of Risk Sharing in Contingent-Claims Market all along with the key concepts of Budget Line, Iso-Expected Value Line, Choices of Decision-makers, Indifference Curves with Risk Averseness. Tutorsglobe offers homework help, assignment help and tutor’s assistance on risk sharing in contingent-claims market.
www.tutorsglobe.com offers Quality Function Deployment (QFD) homework help, assignment help, case study, writing homework help, online tutoring assistance by computer science tutors.
Theory and lecture notes of Independent Samples all along with the key concepts of Independent samples, Sums and Differences of Independent Variables, Difference of the Means, Population Variances Known, Population Variances Unknown. Tutorsglobe offers homework help, assignment help and tutor’s assistance on Independent Samples.
tutorsglobe.com conservation of energy assignment help-homework help by online natural resources tutors
Incompatibility mechanisms tutorial all along with the key concepts of Self-incompatibility in plants, Mechanisms of self-incompatibility, Gametophytic self-incompatibility, Heteromorphic self-incompatibility, Cryptic self-incompatibility, Self-compatibility
Negative Feedback tutorial all along with the key concepts of Benefits of Negative Feedback, Gain Stability, Decreased Distortion, Feedback over Several Stages, Forms of Negative Feedback, Shunt-Derived Series-Fed Voltage Feedback, Current-Series Feedback Amplifier
1950266
Questions Asked
3689
Tutors
1467347
Questions Answered
Start Excelling in your courses, Ask an Expert and get answers for your homework and assignments!!