CMS HB-1 Geometrical check by photogrammetry of the damaged wedge 16 October 03 rd, 2001 CMS HB 1

Size: px
Start display at page:

Download "CMS HB-1 Geometrical check by photogrammetry of the damaged wedge 16 October 03 rd, 2001 CMS HB 1"

Transcription

1 GENEVE, SUISSE From: R. GOUDARD EP/CMI C. HUMBERTCLAUDE EP/CMI J. F. FUCHS EST/SU Date: October 03 rd, 2001 To : A. BALL EP/CMO E. ALVAREZ FCM I. CHURIN Fermilab D. CAMPI EP/CMI G. FABER EP/HC J. FREEMAN Fermilab JP. GIROD EP/HC D. GREEN Fermilab A. HERVE EP/CMI C. LASSEUR EST/SU D.LAZIC EP/HC M. LEBEAU EP/CMA V. POLUBOTKO Fermilab R. SCHMIDT EP/CMO A. SKUJA Fermilab L. VEILLET EP/L3D CMS HB 1 GEOMETRICAL CHECK BY PHOTOGRAMMETRY OF THE DAMAGED WEDGE 16 at CERN SX5 September 2001 This report is available on EDMS /CMS hbm1_w16_4011.doc Page 1 / 22 R. Goudard / C. Humbertclaude / J. F. Fuchs

2

3 I - INTRODUCTION: The wedge W16 of the HCAL BARREL HB-1 fell down on the road the 3 rd of september 2001 during the transportation between CERN site and the SX5 assembly hall. The technical coordinator requested a geometrical check of this wedge before being disassembled and repaired. Following a discusion with I. Churin the 4 th of september we decided to use photogrammetry to measure THREE planes of this wedge : the TOP, the BOTTOM and one SIDE. II THE TEMPERATURE OF THE WEDGE DURING PHOTOGRAMMETRY A temperature sensor was stuck on the wedge during the measurements by photogrammetry. The following graphic shows that temperature of the wedge kept stable within 0.5º during the measurements. The mean temperature was 19.0º (see figure1) 21 Temperature of the wedge W16 during the photogrammetry Measurement with photogrammetry Temperature of the wedge [deg. cel.] /10/01 13:26 9/10/01 13:55 9/10/01 14:24 9/10/01 14:52 9/10/01 15:21 9/10/01 15:50 9/10/01 16:19 9/10/01 16:48 9/10/01 17:16 9/10/01 17:45 Date and time Figure 1: temperature of the wedge during the photogrammetry hbm1_w16_4011.doc Page 3 / 22 R. Goudard / C. Humbertclaude / J. F. Fuchs

4 III - THE SYSTEM FIXING: In agreement with Igor Churin we have chosen the bottom plane as the reference one. The plane [XY] is parallel to the bottom plane. Three well distributed measured points belonging to the bottom plane define the system fixing (see figure 2). The calculation has been done in a nonconstraints system fixing ( six coordinates + 1 distance given by a scale bar). Z Top of the Wedge 16 Bottom of the Wedge 16 Y X Figure 2: The system fixing hbm1_w16_4011.doc Page 4 / 22 R. Goudard / C. Humbertclaude / J. F. Fuchs

5 IV THE MEASURED POINTS: The following pictures show the position and the numbering of the measured points for each side BOTTOM X Figure 3: the bottom plane, reference plane for the calculations, points 90xx TOP Figure 4: the top plane, points 80xx hbm1_w16_4011.doc Page 5 / 22 R. Goudard / C. Humbertclaude / J. F. Fuchs

6 SIDE Figure 5: the side plane, points 70xx hbm1_w16_4011.doc Page 6 / 22 R. Goudard / C. Humbertclaude / J. F. Fuchs

7 V THE RESULTS: 5.1 Accuracy of the measurement: The final least squares bundle adjustment gives 3d coordinates and the RMS error in XYZ for all the points. The scale was established by using independent calibrated carbon Scale Bars distances. VERY IMPORTANT: The given co-ordinates are the co-ordinates of the centres of the survey target. The average RMS (one sigma) achieved for XYZ co-ordinates are: RMS XYZ = 0.2mm 5. 2 Best-fit plane calculation : For each side (Bottom, Top and Side), the best-fit plane has been calculated. Those calculation give the equation of the planes and the distances to the mean plane (see appendix 1) The graphic of the longitudinal and of the transversal profile are provided (see appendix 1) In order to have a better estimation of the shape of each face, several 3d views are presented (see appendix 2) Remark: Distances are signed distances to plane: - Distance with Sign (-) means that the Origin and the Point are on the same side with respect to the Plane - Distance with Sign (+) means that the Origin and the Point are on the opposite side with respect to the Plane Distance signed (+) X or Z Distance signed (+) Origin Y or X Distance signed (-) Distance signed (-) hbm1_w16_4011.doc Page 7 / 22 R. Goudard / C. Humbertclaude / J. F. Fuchs

8 5.3 Summary of the results of the best-fit planes: the bottom plane: The mean plane has been calculated for the BOTTOM side with the 40 points measured by photogrammetry (from 9100 to 9804, please refer to page 5). Equation of the plane * x * y * z (m) = 0 Largest Distance from Plane on + side (mm) 3.0 At Point 9800 Largest Distance from Plane on - side (mm) -2.0 At Point The Top plane: The mean plane has been calculated for the TOP side with the 28 points measured by photogrammetry (from 8100 to 8703, please refer to page 5). Equation of the plane * x * y * z (m) = 0 Largest Distance from Plane on + side (mm) 1.5 At Point 8100 Largest Distance from Plane on - side (mm) -1.4 At Point The Side plane: The mean plane has been calculated for the SIDE with the 132 points measured by photogrammetry (from 7500 to 7773, please refer to page6). Equation of the plane * x * y * z (m) = 0 Largest Distance from Plane on + side (mm) 5.9 At Point 7773 Largest Distance from Plane on - side (mm) -3.2 At Point 7770 hbm1_w16_4011.doc Page 8 / 22 R. Goudard / C. Humbertclaude / J. F. Fuchs

9 APPENDIX 1 : THE BEST-FIT PLANES, BOTTOM, TOP, SIDE BOTTOM side, best fit plane calculation: Results of Plane Fitting - Centroid Method Results_Plane_bottom Equation and Direction Cosines of the Plane : Eqn of a Plane: Z + B*X + C*Y + D = 0 B sig_b mm/m C sig_c mm/m D (m) sig_d mm Hence for Eqn of the form: a*x + b*y + c*z +d = 0 with a, b, c : Dir. Cosines of perp. Line to the Plane a b c d (m) Bearing and Vertical Angle of the Vector from the origin to the plane Bearing (Grades) Vertical Angle (Grades) Dist from the origin to the plane (m) Name Weight Dist. to mean plane [mm] Name Weight Dist. to mean plane [mm] Dist = Signed Dist. to Plane : ( Sign - : Origin & Pt on same side / Plane ) ( Sign + : Origin & Pt on opp. side / Plane ) dx, dy, dz = Diff. co-ordinates : ( Diff. co-ordinates = Pt. proj. - Pt. obs. ) Equation of the plane * x * y * z (m) = 0 Largest Distance from Plane on + side (mm) 3.0 At Point 9800 Largest Distance from Plane on - side (mm) -2.0 At Point 9504 Dist = Signed Dist. to Plane ( - => Origin & Pt on same side / Plane, + => Origin & Pt on opp. side / Plane ) Bearing and Vertical Angle of the Vector perpendicular to the plane Bearing (Grades) Vertical Angle (Grades) hbm1_w16_4011.doc Page 9 / 22 R. Goudard / C. Humbertclaude / J. F. Fuchs

10 4 GRAPHIC 1 : BOTTOM LONGITUDINAL PROFILE Dammaged Wedge HB-1_16, BOTTOM, Distances to mean plane by lines parallel to X axis 3 Dist. to mean plane [mm] to to to to to X [m] Z Top of the Wedge 16 Side of the Wedge 16 Bottom of the Wedge 16 Y X hbm1_w16_4011.doc Page 10 / 22 R. Goudard / C. Humbertclaude / J. F. Fuchs

11 GRAPHIC 2 : BOTTOM TRANSVERSAL PROFILE 4 Dammaged Wedge HB-1_16, BOTTOM, Distances to mean plane by lines parallel to Y axis 3 Dist. to mean plane [mm] xx 92xx 93xx 94xx 95xx 96xx 97xx 98xx Y [m] Z Top of the Wedge 16 Side of the Wedge 16 Bottom of the Wedge 16 Y X hbm1_w16_4011.doc Page 11 / 22 R. Goudard / C. Humbertclaude / J. F. Fuchs

12

13 TOP side, best fit plane calculation: Results of Plane Fitting - Centroid Method Results_Plane_top Equation and Direction Cosines of the Plane : Eqn of a Plane: Z + B*X + C*Y + D = 0 B sig_b mm/m C sig_c mm/m D (m) sig_d mm Hence for Eqn of the form: a*x + b*y + c*z +d = 0 with a, b, c : Dir. Cosines of perp. Line to the Plane a b c d (m) Bearing and Vertical Angle of the Vector from the origin to the plane Bearing (Grades) Vertical Angle (Grades) Dist from the origin to the plane (m) Name Weight Dist. to mean plane [mm] Dist = Signed Dist. to Plane : ( Sign - : Origin & Pt on same side / Plane ) ( Sign + : Origin & Pt on opp. side / Plane ) dx, dy, dz = Diff. co-ordinates ( Diff. co-ordinates = Pt. proj. - Pt. obs. ) Equation of the plane * x * y * z (m) = 0 Largest Distance from Plane on + side (mm) 1.5 At Point 8100 Largest Distance from Plane on - side (mm) -1.4 At Point 8400 Dist = Signed Dist. to Plane ( - => Origin & Pt on same side / Plane, + => Origin & Pt on opp. side / Plane ) Bearing and Vertical Angle of the Vector perpendicular to the plane Bearing (Grades) Vertical Angle (Grades) hbm1_w16_4011.doc Page 13 / 22 R. Goudard / C. Humbertclaude / J. F. Fuchs

14 4 GRAPHIC 3 : TOP LONGITUDINAL PROFILE Dammaged Wedge HB-1_16, TOP, Distances to mean plane by lines parallel to X axis 3 Dist. to mean plane [mm] to to to to X [m] Z Top of the Wedge 16 Side of the Wedge 16 Bottom of the Wedge 16 Y X hbm1_w16_4011.doc Page 14 / 22 R. Goudard / C. Humbertclaude / J. F. Fuchs

15 GRAPHIC 4 : TOP TRANSVERSAL PROFILE 2 Dammaged Wedge HB-1_16, TOP, Distances to mean plane by lines parallel to Y axis 2 Dist. to mean plane [mm] xx 82xx 83xx 84xx 85xx 86xx 87xx Y [m] Z Top of the Wedge 16 Side of the Wedge 16 Bottom of the Wedge 16 Y X hbm1_w16_4011.doc Page 15 / 22 R. Goudard / C. Humbertclaude / J. F. Fuchs

16

17 SIDE best fit plane calculation: Results of Plane Fitting - Centroid Method Results_Plane_side Equation and Direction Cosines of the Plane : Eqn of a Plane: Y + B*Z + C*X + D = 0 B sig_b mm/m C sig_c mm/m D (m) sig_d mm Hence for Eqn of the form: a*x + b*y + c*z +d = 0 with a, b, c : Dir. Cosines of perp. Line to the Plane a b c d (m) Bearing and Vertical Angle of the Vector from the origin to the plane Bearing (Grades) Vertical Angle (Grades) Dist from the origin to the plane (m) Name Weight Dist. to mean plane [mm] Name Weight Dist. to mean plane [mm] hbm1_w16_4011.doc Page 17 / 22 R. Goudard / C. Humbertclaude / J. F. Fuchs

18 Name Weight Dist. to mean plane [mm] Name Weight Dist. to mean plane [mm] Dist = Signed Dist. to Plane : ( Sign - : Origin & Pt on same side / Plane ) ( Sign + : Origin & Pt on opp. side / Plane ) Equation of the plane * x * y * z (m) = 0 Largest Distance from Plane on + side (mm) 5.9 At Point 7773 Largest Distance from Plane on - side (mm) -3.2 At Point 7770 Dist = Signed Dist. to Plane ( - => Origin & Pt on same side / Plane, + => Origin & Pt on opp. side / Plane ) Bearing and Vertical Angle of the Vector perpendicular to the plane Bearing (Grades) Vertical Angle (Grades) hbm1_w16_4011.doc Page 18 / 22 R. Goudard / C. Humbertclaude / J. F. Fuchs

19 GRAPHIC 5 : SIDE LONGITUDINAL PROFILE 6 Dammaged Wedge HB-1_16, SIDE, Distances to mean plane by lines parallel to X axis 4 Dist. to mean plane [mm] x 753x 756x 759x 762x 765x 768x 771x 773x 772x 777x X [m] Z Top of the Wedge 16 Side of the Wedge 16 Y X Bottom of the Wedge 16 hbm1_w16_4011.doc Page 19 / 22 R. Goudard / C. Humbertclaude / J. F. Fuchs

20 GRAPHIC 6 : SIDE TRANSVERSAL PROFILE Dammaged Wedge HB-1_16, SIDE, Distances to mean plane by lines parallel to Z axis 6 4 Dist. to mean plane [mm] to to to to to Z [m] Z Top of the Wedge 16 Side of the Wedge 16 Bottom of the Wedge 16 Y X hbm1_w16_4011.doc Page 20 / 22 R. Goudard / C. Humbertclaude / J. F. Fuchs

21 APPENDIX 2 :THE 3D SURFACES: In order to have a better estimation of the shape of each face, several 3d views are presented : The Bottom plane : 27/09/01 Bottom of the W16 Distances to Mean plane [mm] Vertical Side Column from 9800 to 9804 Column from 9100 to 9104 Inclined Side X [m] R. Goudard, C. Humbertclaude, EC/CMI, JF FUCHS EST/SU The Top plane : 27/09/01 Top of the W16 Column from 8700 to 8703 Distance to Mean plane [mm] Inclined Side Vertical Side Column from 8100 to 8103 X [m] R. Goudard, C. Humbertclaude, EC/CMI, JF FUCHS EST/SU hbm1_w16_4011.doc Page 21 / 22 R. Goudard / C. Humbertclaude / J. F. Fuchs

22 The Side plane : Side of the W16 27/09/01 Inclined Side Distance to Mean plane [mm] Line from 7770 to 7773 Line from 7500 to 7504 X [m] Vertical Side R. Goudard, C. Humbertclaude, EC/CMI, JF FUCHS EST/SU Distance to Mean plane [mm] Inclined Side Side of the W16 27/09/01 Line from 7500 to 7504 Line from 7770 to 7773 X [m] Vertical Side R. Goudard, C. Humbertclaude, EC/CMI, JF FUCHS EST/SU hbm1_w16_4011.doc Page 22 / 22 R. Goudard / C. Humbertclaude / J. F. Fuchs

If the center of the sphere is the origin the the equation is. x y z 2ux 2vy 2wz d 0 -(2)

If the center of the sphere is the origin the the equation is. x y z 2ux 2vy 2wz d 0 -(2) Sphere Definition: A sphere is the locus of a point which remains at a constant distance from a fixed point. The fixed point is called the centre and the constant distance is the radius of the sphere.

More information

High Precision Optical Instrumentation for Large Structures Position Monitoring: The BCAM System Applied to the CMS Magnet

High Precision Optical Instrumentation for Large Structures Position Monitoring: The BCAM System Applied to the CMS Magnet High Precision Optical Instrumentation for Large Structures Position Monitoring: The BCAM System Applied to the CMS Magnet James Bensinger, Kevan Hashemi, Brandeis University P.O. Box 549110 Waltham, MA

More information

1.5 Equations of Lines and Planes in 3-D

1.5 Equations of Lines and Planes in 3-D 1.5. EQUATIONS OF LINES AND PLANES IN 3-D 55 Figure 1.16: Line through P 0 parallel to v 1.5 Equations of Lines and Planes in 3-D Recall that given a point P = (a, b, c), one can draw a vector from the

More information

THREE DIMENSIONAL GEOMETRY

THREE DIMENSIONAL GEOMETRY For more important questions visit : www4onocom CHAPTER 11 THREE DIMENSIONAL GEOMETRY POINTS TO REMEMBER Distance between points P(x 1 ) and Q(x, y, z ) is PQ x x y y z z 1 1 1 (i) The coordinates of point

More information

Class IX Mathematics (Ex. 3.1) Questions

Class IX Mathematics (Ex. 3.1) Questions Class IX Mathematics (Ex. 3.1) Questions 1. How will you describe the position of a table lamp on your study table to another person? 2. (Street Plan): A city has two main roads which cross each other

More information

GEOMETRY IN THREE DIMENSIONS

GEOMETRY IN THREE DIMENSIONS 1 CHAPTER 5. GEOMETRY IN THREE DIMENSIONS 1 INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW GEOMETRY IN THREE DIMENSIONS Contents 1 Geometry in R 3 2 1.1 Lines...............................................

More information

High Precision Optical Instrumentation for Large Structure Position Monitoring: The BCAM System Applied to the CMS Magnet

High Precision Optical Instrumentation for Large Structure Position Monitoring: The BCAM System Applied to the CMS Magnet High Precision Optical Instrumentation for Large Structure Position Monitoring: The BCAM System Applied to the CMS Magnet International Workshop on Accelerator Alignment Stanford Linear Accelerator Center

More information

6. Find the equation of the plane that passes through the point (-1,2,1) and contains the line x = y = z.

6. Find the equation of the plane that passes through the point (-1,2,1) and contains the line x = y = z. Week 1 Worksheet Sections from Thomas 13 th edition: 12.4, 12.5, 12.6, 13.1 1. A plane is a set of points that satisfies an equation of the form c 1 x + c 2 y + c 3 z = c 4. (a) Find any three distinct

More information

Reflections, Translations, and Dilations

Reflections, Translations, and Dilations Reflections, Translations, and Dilations Step 1: Graph and label the following points on your coordinate plane. A (2,2) B (2,8) C (8,8) D (8,2) Step 2: Step 3: Connect the dots in alphabetical order to

More information

WJEC LEVEL 2 CERTIFICATE 9550/01 ADDITIONAL MATHEMATICS

WJEC LEVEL 2 CERTIFICATE 9550/01 ADDITIONAL MATHEMATICS Surname Other Names Centre Number 0 Candidate Number WJEC LEVEL 2 CERTIFICATE 9550/01 ADDITIONAL MATHEMATICS A.M. MONDAY, 24 June 2013 2 1 hours 2 ADDITIONAL MATERIALS A calculator will be required for

More information

1 EquationsofLinesandPlanesin 3-D

1 EquationsofLinesandPlanesin 3-D 1 EquationsofLinesandPlanesin 3-D Recall that given a point P (a, b, c), one can draw a vector from the origin to P. Such a vector is called the position vector of the point P and its coordinates are a,

More information

Updated: March 31, 2016 Calculus III Section Math 232. Calculus III. Brian Veitch Fall 2015 Northern Illinois University

Updated: March 31, 2016 Calculus III Section Math 232. Calculus III. Brian Veitch Fall 2015 Northern Illinois University Updated: March 3, 26 Calculus III Section 5.6 Math 232 Calculus III Brian Veitch Fall 25 Northern Illinois University 5.6 Triple Integrals In order to build up to a triple integral let s start back at

More information

AQA GCSE Further Maths Topic Areas

AQA GCSE Further Maths Topic Areas AQA GCSE Further Maths Topic Areas This document covers all the specific areas of the AQA GCSE Further Maths course, your job is to review all the topic areas, answering the questions if you feel you need

More information

Mathematical Analysis of Tetrahedron (solid angle subtended by any tetrahedron at its vertex)

Mathematical Analysis of Tetrahedron (solid angle subtended by any tetrahedron at its vertex) From the SelectedWorks of Harish Chandra Rajpoot H.C. Rajpoot Winter March 29, 2015 Mathematical Analysis of Tetrahedron solid angle subtended by any tetrahedron at its vertex) Harish Chandra Rajpoot Rajpoot,

More information

Date: 16 July 2016, Saturday Time: 14:00-16:00 STUDENT NO:... Math 102 Calculus II Midterm Exam II Solutions TOTAL. Please Read Carefully:

Date: 16 July 2016, Saturday Time: 14:00-16:00 STUDENT NO:... Math 102 Calculus II Midterm Exam II Solutions TOTAL. Please Read Carefully: Date: 16 July 2016, Saturday Time: 14:00-16:00 NAME:... STUDENT NO:... YOUR DEPARTMENT:... Math 102 Calculus II Midterm Exam II Solutions 1 2 3 4 TOTAL 25 25 25 25 100 Please do not write anything inside

More information

Unit 12 Topics in Analytic Geometry - Classwork

Unit 12 Topics in Analytic Geometry - Classwork Unit 1 Topics in Analytic Geometry - Classwork Back in Unit 7, we delved into the algebra and geometry of lines. We showed that lines can be written in several forms: a) the general form: Ax + By + C =

More information

Math (Spring 2009): Lecture 5 Planes. Parametric equations of curves and lines

Math (Spring 2009): Lecture 5 Planes. Parametric equations of curves and lines Math 18.02 (Spring 2009): Lecture 5 Planes. Parametric equations of curves and lines February 12 Reading Material: From Simmons: 17.1 and 17.2. Last time: Square Systems. Word problem. How many solutions?

More information

14.4: Tangent Planes and Linear Approximations

14.4: Tangent Planes and Linear Approximations 14.4: Tangent Planes and Linear Approximations Marius Ionescu October 15, 2012 Marius Ionescu () 14.4: Tangent Planes and Linear Approximations October 15, 2012 1 / 13 Tangent Planes Marius Ionescu ()

More information

Multivariable Calculus

Multivariable Calculus Multivariable Calculus Chapter 10 Topics in Analytic Geometry (Optional) 1. Inclination of a line p. 5. Circles p. 4 9. Determining Conic Type p. 13. Angle between lines p. 6. Parabolas p. 5 10. Rotation

More information

SIMULGEO AND ITS APPLICATION FOR THE MUON BARREL POSITION MONITOR

SIMULGEO AND ITS APPLICATION FOR THE MUON BARREL POSITION MONITOR 1 SIMULGEO AND ITS APPLICATION FOR THE MUON BARREL POSITION MONITOR Laurent Brunel CERN (CH 1211 Genève 23) and Kossuth University (Debrecen Hungary) 1. INTRODUCTION The design process of the Muon Barrel

More information

1.5 Equations of Lines and Planes in 3-D

1.5 Equations of Lines and Planes in 3-D 56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Figure 1.16: Line through P 0 parallel to v 1.5 Equations of Lines and Planes in 3-D Recall that given a point P = (a, b, c), one can draw a vector from

More information

Three Dimensional Geometry. Linear Programming

Three Dimensional Geometry. Linear Programming Three Dimensional Geometry Linear Programming A plane is determined uniquely if any one of the following is known: The normal to the plane and its distance from the origin is given, i.e. equation of a

More information

Chapter 1. Linear Equations and Straight Lines. 2 of 71. Copyright 2014, 2010, 2007 Pearson Education, Inc.

Chapter 1. Linear Equations and Straight Lines. 2 of 71. Copyright 2014, 2010, 2007 Pearson Education, Inc. Chapter 1 Linear Equations and Straight Lines 2 of 71 Outline 1.1 Coordinate Systems and Graphs 1.4 The Slope of a Straight Line 1.3 The Intersection Point of a Pair of Lines 1.2 Linear Inequalities 1.5

More information

MATH 19520/51 Class 6

MATH 19520/51 Class 6 MATH 19520/51 Class 6 Minh-Tam Trinh University of Chicago 2017-10-06 1 Review partial derivatives. 2 Review equations of planes. 3 Review tangent lines in single-variable calculus. 4 Tangent planes to

More information

Section A1: Gradients of straight lines

Section A1: Gradients of straight lines Time To study this unit will take you about 10 hours. Trying out and evaluating the activities with your pupils in the class will be spread over the weeks you have planned to cover the topic. 31 Section

More information

12 - THREE DIMENSIONAL GEOMETRY Page 1 ( Answers at the end of all questions ) = 2. ( d ) - 3. ^i - 2. ^j c 3. ( d )

12 - THREE DIMENSIONAL GEOMETRY Page 1 ( Answers at the end of all questions ) = 2. ( d ) - 3. ^i - 2. ^j c 3. ( d ) - THREE DIMENSIONAL GEOMETRY Page ( ) If the angle θ between the line x - y + x + y - z - and the plane λ x + 4 0 is such that sin θ, then the value of λ is - 4-4 [ AIEEE 00 ] ( ) If the plane ax - ay

More information

Section 13.5: Equations of Lines and Planes. 1 Objectives. 2 Assignments. 3 Lecture Notes

Section 13.5: Equations of Lines and Planes. 1 Objectives. 2 Assignments. 3 Lecture Notes Section 13.5: Equations of Lines and Planes 1 Objectives 1. Find vector, symmetric, or parametric equations for a line in space given two points on the line, given a point on the line and a vector parallel

More information

Contents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Homework 1 - Solutions 3. 2 Homework 2 - Solutions 13

Contents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Homework 1 - Solutions 3. 2 Homework 2 - Solutions 13 MATH 32B-2 (8) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables Contents Homework - Solutions 3 2 Homework 2 - Solutions 3 3 Homework 3 - Solutions 9 MATH 32B-2 (8) (L) G. Liu / (TA) A. Zhou Calculus

More information

Conic Sections. College Algebra

Conic Sections. College Algebra Conic Sections College Algebra Conic Sections A conic section, or conic, is a shape resulting from intersecting a right circular cone with a plane. The angle at which the plane intersects the cone determines

More information

CSE452 Computer Graphics

CSE452 Computer Graphics CSE45 Computer Graphics Lecture 8: Computer Projection CSE45 Lecture 8: Computer Projection 1 Review In the last lecture We set up a Virtual Camera Position Orientation Clipping planes Viewing angles Orthographic/Perspective

More information

Analytical Solid Geometry

Analytical Solid Geometry Analytical Solid Geometry Distance formula(without proof) Division Formula Direction cosines Direction ratios Planes Straight lines Books Higher Engineering Mathematics by B S Grewal Higher Engineering

More information

2) In a right triangle, with acute angle θ, sin θ = 7/9. What is the value of tan θ?

2) In a right triangle, with acute angle θ, sin θ = 7/9. What is the value of tan θ? CC Geometry H Aim #26: Students rewrite the Pythagorean theorem in terms of sine and cosine ratios and write tangent as an identity in terms of sine and cosine. Do Now: 1) In a right triangle, with acute

More information

If three points A (h, 0), P (a, b) and B (0, k) lie on a line, show that: a b 1.

If three points A (h, 0), P (a, b) and B (0, k) lie on a line, show that: a b 1. ASSIGNMENT ON STRAIGHT LINES LEVEL 1 (CBSE/NCERT/STATE BOARDS) 1 Find the angle between the lines joining the points (0, 0), (2, 3) and the points (2, 2), (3, 5). 2 What is the value of y so that the line

More information

MATH 31A HOMEWORK 9 (DUE 12/6) PARTS (A) AND (B) SECTION 5.4. f(x) = x + 1 x 2 + 9, F (7) = 0

MATH 31A HOMEWORK 9 (DUE 12/6) PARTS (A) AND (B) SECTION 5.4. f(x) = x + 1 x 2 + 9, F (7) = 0 FROM ROGAWSKI S CALCULUS (2ND ED.) SECTION 5.4 18.) Express the antiderivative F (x) of f(x) satisfying the given initial condition as an integral. f(x) = x + 1 x 2 + 9, F (7) = 28.) Find G (1), where

More information

Data handling 2: Transformations

Data handling 2: Transformations Intro Geo information Science (GRS 10306) Data handling 2: Transformations 2009/2010 CGI GIRS Transformation definition Query a data handling class of operators which doesn t change the thematic and geometric

More information

1. Use the Trapezium Rule with five ordinates to find an approximate value for the integral

1. Use the Trapezium Rule with five ordinates to find an approximate value for the integral 1. Use the Trapezium Rule with five ordinates to find an approximate value for the integral Show your working and give your answer correct to three decimal places. 2 2.5 3 3.5 4 When When When When When

More information

16. LECTURE 16. I understand how to find the rate of change in any direction. I understand in what direction the maximum rate of change happens.

16. LECTURE 16. I understand how to find the rate of change in any direction. I understand in what direction the maximum rate of change happens. 6. LETURE 6 Objectives I understand how to find the rate of change in any direction. I understand in what direction the maximum rate of change happens. So far, we ve learned the definition of the gradient

More information

Curves, Tangent Planes, and Differentials ( ) Feb. 26, 2012 (Sun) Lecture 9. Partial Derivatives: Signs on Level Curves, Tangent

Curves, Tangent Planes, and Differentials ( ) Feb. 26, 2012 (Sun) Lecture 9. Partial Derivatives: Signs on Level Curves, Tangent Lecture 9. Partial Derivatives: Signs on Level Curves, Tangent Planes, and Differentials ( 11.3-11.4) Feb. 26, 2012 (Sun) Signs of Partial Derivatives on Level Curves Level curves are shown for a function

More information

What you will learn today

What you will learn today What you will learn today Tangent Planes and Linear Approximation and the Gradient Vector Vector Functions 1/21 Recall in one-variable calculus, as we zoom in toward a point on a curve, the graph becomes

More information

Tangent Planes and Linear Approximations

Tangent Planes and Linear Approximations February 21, 2007 Tangent Planes Tangent Planes Let S be a surface with equation z = f (x, y). Tangent Planes Let S be a surface with equation z = f (x, y). Let P(x 0, y 0, z 0 ) be a point on S. Tangent

More information

COORDINATE TRANSFORMATION. Lecture 6

COORDINATE TRANSFORMATION. Lecture 6 COORDINATE TRANSFORMATION Lecture 6 SGU 1053 SURVEY COMPUTATION 1 Introduction Geomatic professional are mostly confronted in their work with transformations from one two/three-dimensional coordinate system

More information

S56 (5.3) Higher Straight Line.notebook June 22, 2015

S56 (5.3) Higher Straight Line.notebook June 22, 2015 Daily Practice 5.6.2015 Q1. Simplify Q2. Evaluate L.I: Today we will be revising over our knowledge of the straight line. Q3. Write in completed square form x 2 + 4x + 7 Q4. State the equation of the line

More information

Class IX Chapter 11 Constructions Maths

Class IX Chapter 11 Constructions Maths 1 Class IX Chapter 11 Constructions Maths 1: Exercise 11.1 Question Construct an angle of 90 at the initial point of a given ray and justify the construction. Answer: The below given steps will be followed

More information

Chapter 15: Functions of Several Variables

Chapter 15: Functions of Several Variables Chapter 15: Functions of Several Variables Section 15.1 Elementary Examples a. Notation: Two Variables b. Example c. Notation: Three Variables d. Functions of Several Variables e. Examples from the Sciences

More information

Lecture 17. ENGR-1100 Introduction to Engineering Analysis CENTROID OF COMPOSITE AREAS

Lecture 17. ENGR-1100 Introduction to Engineering Analysis CENTROID OF COMPOSITE AREAS ENGR-00 Introduction to Engineering Analysis Lecture 7 CENTROID OF COMPOSITE AREAS Today s Objective : Students will: a) Understand the concept of centroid. b) Be able to determine the location of the

More information

Section III: TRANSFORMATIONS

Section III: TRANSFORMATIONS Section III: TRANSFORMATIONS in 2-D 2D TRANSFORMATIONS AND MATRICES Representation of Points: 2 x 1 matrix: X Y General Problem: [B] = [T] [A] [T] represents a generic operator to be applied to the points

More information

HP-35s Calculator Program Curves 2A

HP-35s Calculator Program Curves 2A Programmer: Dr. Bill Hazelton Date: March, 2008. Version: 1.0 Mnemonic: P for Parabolic Vertical Curve. Line Instruction Display User Instructions P001 LBL P LBL P P002 CLSTK CLEAR 5 P003 FS? 10 FLAGS

More information

MA 243 Calculus III Fall Assignment 1. Reading assignments are found in James Stewart s Calculus (Early Transcendentals)

MA 243 Calculus III Fall Assignment 1. Reading assignments are found in James Stewart s Calculus (Early Transcendentals) MA 43 Calculus III Fall 8 Dr. E. Jacobs Assignments Reading assignments are found in James Stewart s Calculus (Early Transcendentals) Assignment. Spheres and Other Surfaces Read. -. and.6 Section./Problems

More information

ORIENTATIONS OF LINES AND PLANES IN SPACE

ORIENTATIONS OF LINES AND PLANES IN SPACE GG303 Lab 1 8/21/09 1 ORIENTATIONS OF LINES AND PLANES IN SPACE I Main Topics A Definitions of points, lines, and planes B Geologic methods for describing lines and planes C Attitude symbols for geologic

More information

UNIT 2 2D TRANSFORMATIONS

UNIT 2 2D TRANSFORMATIONS UNIT 2 2D TRANSFORMATIONS Introduction With the procedures for displaying output primitives and their attributes, we can create variety of pictures and graphs. In many applications, there is also a need

More information

Use of Number Maths Statement Code no: 1 Student: Class: At Junior Certificate level the student can: Apply the knowledge and skills necessary to perf

Use of Number Maths Statement Code no: 1 Student: Class: At Junior Certificate level the student can: Apply the knowledge and skills necessary to perf Use of Number Statement Code no: 1 Apply the knowledge and skills necessary to perform mathematical calculations 1 Recognise simple fractions, for example 1 /4, 1 /2, 3 /4 shown in picture or numerical

More information

Graded Assignment 2 Maple plots

Graded Assignment 2 Maple plots Graded Assignment 2 Maple plots The Maple part of the assignment is to plot the graphs corresponding to the following problems. I ll note some syntax here to get you started see tutorials for more. Problem

More information

MATH 2400: CALCULUS 3 MAY 9, 2007 FINAL EXAM

MATH 2400: CALCULUS 3 MAY 9, 2007 FINAL EXAM MATH 4: CALCULUS 3 MAY 9, 7 FINAL EXAM I have neither given nor received aid on this exam. Name: 1 E. Kim................ (9am) E. Angel.............(1am) 3 I. Mishev............ (11am) 4 M. Daniel...........

More information

Multipath Clamp-On Acoustic Flow Meter. Ing. Jürgen Skripalle HydroVision GmbH, Germany

Multipath Clamp-On Acoustic Flow Meter. Ing. Jürgen Skripalle HydroVision GmbH, Germany Dr.-Ing Ing. Jürgen Skripalle HydroVision GmbH, Germany Content Motivation Project Kopswerk II / Austria Signal Coupling Path Angle / Wall Thickness Optical 3D-System / Determination of Positions Discharge

More information

The base of a solid is the region in the first quadrant bounded above by the line y = 2, below by

The base of a solid is the region in the first quadrant bounded above by the line y = 2, below by Chapter 7 1) (AB/BC, calculator) The base of a solid is the region in the first quadrant bounded above by the line y =, below by y sin 1 x, and to the right by the line x = 1. For this solid, each cross-section

More information

Buds Public School, Dubai

Buds Public School, Dubai Buds Public School, Dubai Subject: Maths Grade: 11 AB Topic: Statistics, Probability, Trigonometry, 3D, Conic Section, Straight lines and Limits and Derivatives Statistics and Probability: 1. Find the

More information

PART A (5x5M =25M) dx +2xy 4x2 = 0 and passing through the origin. using the method of multipliers. PART B (5x10M = 50M)

PART A (5x5M =25M) dx +2xy 4x2 = 0 and passing through the origin. using the method of multipliers. PART B (5x10M = 50M) FIRST YEAR B.SC. MATHEMATICS PAPER I SEMESTER I DIFFERENTIAL EQUATIONS MODEL QUESTION PAPER (THEORY) Time: 3 Hours Max. Marks: 75 *This Paper Csists of Two parts. Follow the Instructis Carefully PART A

More information

Rectangular Coordinates in Space

Rectangular Coordinates in Space Rectangular Coordinates in Space Philippe B. Laval KSU Today Philippe B. Laval (KSU) Rectangular Coordinates in Space Today 1 / 11 Introduction We quickly review one and two-dimensional spaces and then

More information

f (Pijk ) V. may form the Riemann sum: . Definition. The triple integral of f over the rectangular box B is defined to f (x, y, z) dv = lim

f (Pijk ) V. may form the Riemann sum: . Definition. The triple integral of f over the rectangular box B is defined to f (x, y, z) dv = lim Chapter 14 Multiple Integrals..1 Double Integrals, Iterated Integrals, Cross-sections.2 Double Integrals over more general regions, Definition, Evaluation of Double Integrals, Properties of Double Integrals.3

More information

Section 7.2 Volume: The Disk Method

Section 7.2 Volume: The Disk Method Section 7. Volume: The Disk Method White Board Challenge Find the volume of the following cylinder: No Calculator 6 ft 1 ft V 3 1 108 339.9 ft 3 White Board Challenge Calculate the volume V of the solid

More information

Moore Catholic High School Math Department

Moore Catholic High School Math Department Moore Catholic High School Math Department Geometry Vocabulary The following is a list of terms and properties which are necessary for success in a Geometry class. You will be tested on these terms during

More information

Centre for Digital Image Measurement and Analysis, School of Engineering, City University, Northampton Square, London, ECIV OHB

Centre for Digital Image Measurement and Analysis, School of Engineering, City University, Northampton Square, London, ECIV OHB HIGH ACCURACY 3-D MEASUREMENT USING MULTIPLE CAMERA VIEWS T.A. Clarke, T.J. Ellis, & S. Robson. High accuracy measurement of industrially produced objects is becoming increasingly important. The techniques

More information

Following are the solutions for the Shop Math Quiz found in the September issue of Tooling & Production magazine s Shop Talk with Steve Rose.

Following are the solutions for the Shop Math Quiz found in the September issue of Tooling & Production magazine s Shop Talk with Steve Rose. RTSI Quiz Solutions Following are the solutions for the Shop Math Quiz found in the September issue of Tooling & Production magazine s Shop Talk with Steve Rose. 1. What is the decimal equivalent of 27/40?

More information

[Anton, pp , pp ] & [Bourne, pp ]

[Anton, pp , pp ] & [Bourne, pp ] hapter 3 Integral Theorems [Anton, pp. 1124 1130, pp. 1145 1160] & [Bourne, pp. 195 224] First of all some definitions which we will need in the following: Definition 3.1. (a) A domain (region) is an open

More information

GCSE LINKED PAIR PILOT 4363/02 METHODS IN MATHEMATICS UNIT 1: Methods (Non-Calculator) HIGHER TIER

GCSE LINKED PAIR PILOT 4363/02 METHODS IN MATHEMATICS UNIT 1: Methods (Non-Calculator) HIGHER TIER Surname Other Names Centre Number 0 Candidate Number GCSE LINKED PAIR PILOT 4363/02 METHODS IN MATHEMATICS UNIT 1: Methods (Non-Calculator) HIGHER TIER A.M. TUESDAY, 11 June 2013 2 hours CALCULATORS ARE

More information

Intro. To Graphing Linear Equations

Intro. To Graphing Linear Equations Intro. To Graphing Linear Equations The Coordinate Plane A. The coordinate plane has 4 quadrants. B. Each point in the coordinate plain has an x-coordinate (the abscissa) and a y-coordinate (the ordinate).

More information

MATHEMATICS (SYLLABUS D) 4024/02

MATHEMATICS (SYLLABUS D) 4024/02 CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Ordinary Level MATHEMATICS (SYLLABUS D) 4024/02 Paper 2 Additional Materials: Answer Booklet/Paper Electronic calculator Geometrical

More information

[HALL PROBE GRADIOMETRY ]

[HALL PROBE GRADIOMETRY ] 2008 [HALL PROBE GRADIOMETRY ] A novel Scanning Hall probe gradiometer has been developed and a new method to image x, y & z components of the magnetic field on the sample surface has been demonstrated

More information

Module 3: Stand Up Conics

Module 3: Stand Up Conics MATH55 Module 3: Stand Up Conics Main Math concepts: Conic Sections (i.e. Parabolas, Ellipses, Hyperbolas), nd degree equations Auxilliary ideas: Analytic vs. Co-ordinate-free Geometry, Parameters, Calculus.

More information

HIE ISOLDE ALIGNMENT AND MONITORING SYSTEM TECHNICAL DESIGN AND PROJECT STATUS

HIE ISOLDE ALIGNMENT AND MONITORING SYSTEM TECHNICAL DESIGN AND PROJECT STATUS IWAA 2012 - Fermilab September 2012 HIE ISOLDE ALIGNMENT AND MONITORING SYSTEM TECHNICAL DESIGN AND PROJECT STATUS Jean-Christophe Gayde Guillaume Kautzmann Sebastian Waniorek BE/ABP-SU BE/ABP-SU BE/ABP-SU

More information

Math 253, Section 102, Fall 2006 Practice Final Solutions

Math 253, Section 102, Fall 2006 Practice Final Solutions Math 253, Section 102, Fall 2006 Practice Final Solutions 1 2 1. Determine whether the two lines L 1 and L 2 described below intersect. If yes, find the point of intersection. If not, say whether they

More information

Exercise (3.1) Question 1: How will you describe the position of a table lamp on your study table to another person?

Exercise (3.1) Question 1: How will you describe the position of a table lamp on your study table to another person? Class IX - NCERT Maths Exercise (3.1) Question 1: How will you describe the position of a table lamp on your study table to another person? Solution 1: Let us consider the given below figure of a study

More information

MEASURES OF CENTRAL TENDENCY AND MEDIAN

MEASURES OF CENTRAL TENDENCY AND MEDIAN I. Solve the following: MEASURES OF CENTRAL TENDENCY AND MEDIAN 1. Find the value of p for the following distribution whose mean is 10. Variate 5 7 9 11 13 15 20 Frequency 4 4 p 7 3 2 1 Grade X Mathematics

More information

Chapter 15 Notes, Stewart 7e

Chapter 15 Notes, Stewart 7e Contents 15.2 Iterated Integrals..................................... 2 15.3 Double Integrals over General Regions......................... 5 15.4 Double Integrals in Polar Coordinates..........................

More information

Math 213 Calculus III Practice Exam 2 Solutions Fall 2002

Math 213 Calculus III Practice Exam 2 Solutions Fall 2002 Math 13 Calculus III Practice Exam Solutions Fall 00 1. Let g(x, y, z) = e (x+y) + z (x + y). (a) What is the instantaneous rate of change of g at the point (,, 1) in the direction of the origin? We want

More information

REVIEW I MATH 254 Calculus IV. Exam I (Friday, April 29) will cover sections

REVIEW I MATH 254 Calculus IV. Exam I (Friday, April 29) will cover sections REVIEW I MATH 254 Calculus IV Exam I (Friday, April 29 will cover sections 14.1-8. 1. Functions of multivariables The definition of multivariable functions is similar to that of functions of one variable.

More information

Algebra 2 Honors Lesson 10 Translating Functions

Algebra 2 Honors Lesson 10 Translating Functions Algebra 2 Honors Lesson 10 Translating Functions Objectives: The students will be able to translate a base function horizontally and vertically. Students will be able to describe the translation of f(x)

More information

P1 REVISION EXERCISE: 1

P1 REVISION EXERCISE: 1 P1 REVISION EXERCISE: 1 1. Solve the simultaneous equations: x + y = x +y = 11. For what values of p does the equation px +4x +(p 3) = 0 have equal roots? 3. Solve the equation 3 x 1 =7. Give your answer

More information

Grade IX. Mathematics Geometry Notes. #GrowWithGreen

Grade IX. Mathematics Geometry Notes. #GrowWithGreen Grade IX Mathematics Geometry Notes #GrowWithGreen The distance of a point from the y - axis is called its x -coordinate, or abscissa, and the distance of the point from the x -axis is called its y-coordinate,

More information

Ray Tracing II 2. Ray Tracing with Ray

Ray Tracing II 2. Ray Tracing with Ray Ray Tracing II Astronomy 6525 Lecture 3: Part 2 Ray Tracing with Ray Ray tracing with Ray Introduction to Ray Using the program Worked example HBO example Ray Tracing II 2 1 Starting ray Entering the prescription

More information

Co-ordinate Geometry

Co-ordinate Geometry Co-ordinate Geometry 1. Find the value of P for which the points (1, -), (2, -6) and (p, -1) are collinear 2. If the point P (x, y) is equidistant from the points A (1,) and B(4, 1). Prove that 2x+y =

More information

Computational Geometry Lab: SEARCHING A TETRAHEDRAL MESH

Computational Geometry Lab: SEARCHING A TETRAHEDRAL MESH Computational Geometry Lab: SEARCHING A TETRAHEDRAL MESH John Burkardt Information Technology Department Virginia Tech http://people.sc.fsu.edu/ jburkardt/presentations/cg lab search tet mesh.pdf December

More information

CHAPTER - 10 STRAIGHT LINES Slope or gradient of a line is defined as m = tan, ( 90 ), where is angle which the line makes with positive direction of x-axis measured in anticlockwise direction, 0 < 180

More information

Ray Tracing I. History

Ray Tracing I. History History Ray Tracing came from the Physics of lens making. The process was that of drawing lines or rays through a glass shape to determine it s lens properties. It is also related to early perspective

More information

1. Each interior angle of a polygon is 135. How many sides does it have? askiitians

1. Each interior angle of a polygon is 135. How many sides does it have? askiitians Class: VIII Subject: Mathematics Topic: Practical Geometry No. of Questions: 19 1. Each interior angle of a polygon is 135. How many sides does it have? (A) 10 (B) 8 (C) 6 (D) 5 (B) Interior angle =. 135

More information

Math 326 Assignment 3. Due Wednesday, October 17, 2012.

Math 326 Assignment 3. Due Wednesday, October 17, 2012. Math 36 Assignment 3. Due Wednesday, October 7, 0. Recall that if G(x, y, z) is a function with continuous partial derivatives, and if the partial derivatives of G are not all zero at some point (x 0,y

More information

Geometry: Conic Sections

Geometry: Conic Sections Conic Sections Introduction When a right circular cone is intersected by a plane, as in figure 1 below, a family of four types of curves results. Because of their relationship to the cone, they are called

More information

5200/7200 Fall 2007 Concurrence theorems for triangles

5200/7200 Fall 2007 Concurrence theorems for triangles 5200/7200 Fall 2007 Concurrence theorems for triangles There are two basic concurrence theorems for triangles that hold in neutral geometry, that of medians and of angle bisectors, but it seems hard to

More information

PERFORMANCE TEST OF THE CMS LINK ALIGNMENT SYSTEM

PERFORMANCE TEST OF THE CMS LINK ALIGNMENT SYSTEM PERFORMANCE TEST OF THE CMS LINK ALIGNMENT SYSTEM I. Vila, A.L. Virto Instituto de Física de Cantabria (CSIC- Universidad de Cantabria) Santander, Spain 1. INTRODUCTION The CMS [1] alignment system [2]

More information

Gradient and Directional Derivatives

Gradient and Directional Derivatives Gradient and Directional Derivatives MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011 Background Given z = f (x, y) we understand that f : gives the rate of change of z in

More information

CARTESIAN OVOIDS. A. Cartesian Ovoids in the case n o < n i

CARTESIAN OVOIDS. A. Cartesian Ovoids in the case n o < n i CARTESIAN OVOIDS The Cartesian ovoids are plane curves that have been first described by Rene Descartes (in 1637 AD) and represent the geometric locus of points that have the same linear combination of

More information

1 William is drawing pictures of cross sections of the right circular cone below.

1 William is drawing pictures of cross sections of the right circular cone below. 1 William is drawing pictures of cross sections of the right circular cone below. Which drawing can not be a cross section of a cone? 1) 2) 3) 4) 2 An equation of a line perpendicular to the line represented

More information

with slopes m 1 and m 2 ), if and only if its coordinates satisfy the equation y y 0 = 0 and Ax + By + C 2

with slopes m 1 and m 2 ), if and only if its coordinates satisfy the equation y y 0 = 0 and Ax + By + C 2 CHAPTER 10 Straight lines Learning Objectives (i) Slope (m) of a non-vertical line passing through the points (x 1 ) is given by (ii) If a line makes an angle α with the positive direction of x-axis, then

More information

Wednesday 18 May 2016 Morning

Wednesday 18 May 2016 Morning Oxford Cambridge and RSA Wednesday 18 May 016 Morning AS GCE MATHEMATICS (MEI) 4751/01 Introduction to Advanced Mathematics (C1) QUESTION PAPER * 6 8 8 5 4 5 4 4 * Candidates answer on the Printed Answer

More information

VELO Module Production - Final Module Metrology

VELO Module Production - Final Module Metrology LHCb-2007-087 11 January 2008 VELO Module Production - Final Module Metrology LHCB Technical Note Issue: Draft Revision: 1 Reference: LHCb 2007-087 Created: 10 th October 2006 Last modified: 11 th January

More information

Lines and Planes in 3D

Lines and Planes in 3D Lines and Planes in 3D Philippe B. Laval KSU January 28, 2013 Philippe B. Laval (KSU) Lines and Planes in 3D January 28, 2013 1 / 20 Introduction Recall that given a point P = (a, b, c), one can draw a

More information

Lesson 1. Unit 2 Practice Problems. Problem 2. Problem 1. Solution 1, 4, 5. Solution. Problem 3

Lesson 1. Unit 2 Practice Problems. Problem 2. Problem 1. Solution 1, 4, 5. Solution. Problem 3 Unit 2 Practice Problems Lesson 1 Problem 1 Rectangle measures 12 cm by 3 cm. Rectangle is a scaled copy of Rectangle. Select all of the measurement pairs that could be the dimensions of Rectangle. 1.

More information

Sections through assemblies

Sections through assemblies Sections through assemblies Pay attention to lining Pay attention to representation Pay attention to representation Pay attention to representation Section in a flange AUXILIARY VIEWS Definitions Any view

More information

2 CUTTING TOOL GEOMETRY PRODUCTION TECNOLOGY

2 CUTTING TOOL GEOMETRY PRODUCTION TECNOLOGY 2 CUTTING TOOL GEOMETRY PRODUCTION TECNOLOGY Instructional objectives At the end of this lesson, the student should be able to : (a) conceive rake angle and clearance angle of cutting tools (b) classify

More information

Let s write this out as an explicit equation. Suppose that the point P 0 = (x 0, y 0, z 0 ), P = (x, y, z) and n = (A, B, C).

Let s write this out as an explicit equation. Suppose that the point P 0 = (x 0, y 0, z 0 ), P = (x, y, z) and n = (A, B, C). 4. Planes and distances How do we represent a plane Π in R 3? In fact the best way to specify a plane is to give a normal vector n to the plane and a point P 0 on the plane. Then if we are given any point

More information