DTECH-111-GEOMETRICAL-CONSTRUCTION.pdf

DTech 111 FUNDAMENTALS OF MECHANICAL DRAFTING WITH CAD GEOMETRICAL CONSTRUCTION CREDITED TO: Drawing 111 Workbook: Fundamentals of Technical Sketching and Drawing by Jivulter Mangubat and Marde Ponce (Mutya Publishing House, Inc.) https://www.iitg.ac.in/rkbc/ME111/Lecture2%20Geometrical%20constr...

DTech 111 FUNDAMENTALS OF MECHANICAL DRAFTING WITH CAD GEOMETRICAL CONSTRUCTION CREDITED TO: Drawing 111 Workbook: Fundamentals of Technical Sketching and Drawing by Jivulter Mangubat and Marde Ponce (Mutya Publishing House, Inc.) https://www.iitg.ac.in/rkbc/ME111/Lecture2%20Geometrical%20construction.pdf https://www.cuemath.com/geometry/essence-of-geometrical-constructions/ https://www.slideshare.net/hermiraguilar/engineering-drawing-geometric-construction-lesson-4 https://www.youtube.com/watch?v=hpsx5wpdu0Y  GEOMETRICAL CONSTRUCTION 1 Essence of Geometrical Constructions Geometry is a concept that deals with lines, angles, shapes of objects, sizes and dimensions. To represent shapes on a paper we need to draw them accurately with the help of tools like rulers, protractors or compasses. The simplest construction is that of a line. Geometrical Construction Definition Geometrical construction means drawing lines, line segments, shapes, circles and other figures accurately using a ruler, a compass, or a protractor. Most of the geometrical figures involve drawing a line segment, drawing parallel and perpendicular lines, perpendicular bisectors, circles, and even drawing tangents to circles. Geometrical construction 1 is very crucial in producing quality output in terms in bisecting, trisecting, perpendicular lines, parallel lines, triangles, square and regular polygons are part /element of most common objects. To be truly proficient in the layout of both simple and complex drawing, the student must know and full understand the many geometrical construction method used. Geometrical Entities are as follows: 1. Point: A point describes a position or a location. There is no specific size for a point. A point is represented in technical drawing as a small cross made of dashes that are approximately 3 mm long. 2. Line: A line is a figure with only length and no width. It extends infinitely in both directions. A geometric primitive that has length and direction, but no thickness. It may be straight, curved or a combination of these. Lines also have important relationship or conditions, such as parallel, intersecting, and tangent Straight line - a progression of points where considered as the shortest distance from one point to another Horizontal line - a line along the point of position to 0° Vertical line - a line projected at 90° from the base or horizon Diagonal line - an inclined line (oblique) directed to either left or right Curved line - a progression of points without definite direction of flow Perpendicular line - two line that intersect at 90° with each other Parallel line - two lines that never intersect thru infinity 3. Arc: A curved part is generally described as an arc. Arc is also a part of the circle. A geometrical entity where contains progression of points with equal distance from a common vertex. Minor arc - an arc with angle less than 180° Quarter arc - an arc equal to 90° Major arc - an arc with angles more than 180° but less than 360° Semi-circle - an arc equal to 180° 4. Angle - a geometrical entity where two lines radiating from a common vertex to different direction. Acute angle - an angle that measures more than 0° and not more than 90° Obtuse angle - an angle that measures more than 90° but less than 180° Straight angle - an angle that measures exactly 180°, the lines form a straight line Reflex angle - an angle measures more than 180° but less than 360° Complimentary angle - these are two angle with a sum of 90° Supplementary angle - these are two angles with a sum of 180° Conjugate angle - these are two angles with a sum of 360° Geometrical Planes 1. Triangle - three sided plane figure where the sum of the internal angles of any flat triangle is always 180°. Equilateral triangle - all sides and angles are equal Isosceles triangle - two side equal in length & two angles equal as well Scalene triangle - all side in different length & has unequal angles Right-angle triangle - a triangle which contains one angle of 90° Acute-angle triangle - a triangle which contains three acute angles Obtuse-angle triangle - a triangle which contains one angle over 90° Isosceles-right triangle - a triangle which contains one angle at 90° & two of its side equal in length and angles 2. Quadrilateral - is a four sided plane figure Square - all the sides at the same length & all angles at 90° Rectangular - opposite sides parallel and all the angles at 90° Rhombus - all sides equal but no angles equal to 90° Parallelogram (rhomboid) - has opposite sides parallel to each other but no angles 90° Trapezoid - one of the opposite side parallel Trapezium - no edge parallel and no sides equal 3. Circle - a plane bounded by close progression of points having equal distance from a common vertex called center point. Circumference - Circumference is the distance around the outside of a circle. Another way of describing it is the perimeter of a circle. Chord - A chord is a straight line that joins two points on the circumference. these two points can be anywhere on the circumference. Diameter - A diameter is a chord that passes through the centre point of the circle. A diameter is the longest possible chord you can draw in the circle. Radius - A radius is a line joining the centre of the circle to a point on the circumference. You maybe often see it shortened in drawings to the letter 'R'. For example R50 meaning a radius of 50mm. The plural of radius is radii. Arc - An arc can be any part of the circumference. Quadrant - A quadrant is a quarter of a full circle. Sector - A sector is the area of a circle enclosed by two radii and an arc. A quadrant is also a form of sector. An easy way to remember what a sector is, that it looks like slices of pizza. Segment - A segment is the area of the circle enclosed by a chord and an arc. An easy way to remember what a segment is, that it takes the shape of an orange segment. Radius - A semi-circle is exactly half of a circle. A semi-circle has to contain the diameter of that circle. 4. Ellipse - is created by a point moving along a path where the sum of its distances from two points, each called a focus of an ellipse (foci is the plural form), is equal to the major diameter. 5. Regular Polygon - a plane figure which the length of the sides are equal and are perpendicular to the center of the circle Triangle - three sided polygon, with equal internal angles of 60° with a sum of 180° Square - a four-sided polygon, equal internal angles of 90° with a sum of 360° Pentagon - five-sided polygon with an equal internal angle of 108° & a sum of 540° Hexagon - six-sided polygon with an equal internal angle of 120° & a sum of 720° Heptagon - seven-sided polygon with an equal internal angle of 128.6° & a sum of 900° Octagon - eight-sided polygon with an equal internal angle of 135° & a sum of 1080° Nonagon - nine-sided polygon with an equal internal angle of 140° & a sum of 1260° Decagon - ten-sided polygon with an equal internal angle of 144° & a sum of 1440° Hendecagon - eleven-sided polygon with an equal internal angle of 147.3°& a sum of 1620° Dodecagon - twelve-sided polygon with an equal internal angle of 150° & a sum of 1800° Geometrical Solids - a figure which contains volume and are representations of real world objects. A 3 dimensional, they have length, width, & depth. A. Primitives - the basic solids widely used in creating and representing of elementary forms of common object. Cube - has six equal faces, each of which is a square Prism - any polygon/closed entities (other than circle or ellipse) extruded to a certain thickness. Cone - has circular/elliptical base projected to a certain apex point oblique/inclined or perpendicular from the base Cylinder - has circular/elliptical base projected to a thickness oblique/inclined or perpendicular from the base Pyramid - a solid having a polygonal base and the triangular sides are projected to an apex point, oblique/inclined or perpendicular from the base Sphere - a globe-like solid having equal distance from the center of any surface of the solid Torus - a circular tube-like solid enclosing a circle path Ellipsoid - a solid formed from different distance of two opposite perpendicular axes (ellipse) from the center of the object. Maybe a Prolate or an Oblate  Methods of Geometrical Construction 1. Bisecting - to bisect means to divide an entity into 2 equal partitions. This method can also be applied to divisions o 4, 8, 16, 32, & forth. A. Bisecting a Line or an Arc (Compass method) B. Bisecting a Line or an Arc (Triangle From A & B draw equal arcs with radius method) greater than half AB. Then join the From endpoints A & B, draw construction lines intersections D & E with a straight line to at 30, 45, or 60 degrees with the given line. Then locate the midpoint. through their intersection, C, draw a line perpendicular to the given line to locate the center C C. Dividing a Line into equal proportion (Compass D. Dividing a Line into equal proportion (Triangle method) method) Apply the method of bisecting a line using From endpoints A & B, draw construction lines compass to achieve 2,4,8,16,36, etc. partitions. at 30, 45, or 60 degrees with the given line. Then through their intersection, C, draw a line perpendicular to the given line to locate the center C E. Bisecting an Angle F. Transfer an Angle to new location Angle BAC is to be bisected „ Strike large arc Angle BAC is to be transferred to the new R. From intersection points C & B, strike equal position A' B'. Use any convenient radius R, arcs r with radius slightly larger than half BC, and strike arcs from centers A and A'. Strike to intersect at D. Draw line AD, which bisect equal arcs r, and draw side A' C'. the angle. 2. Trisecting - to trisect means to divide into 3. Perpendicular lines - lines having an three equal partitions. This method can be applies intersection of 90° angle. to any number of partitions. a. Erecting a perpendicular line. Trisecting a line is simply projecting a line from one Prolong the line segment given to serve as a pivot of the endpoints of the given line to any direction. of a bisecting procedure then from the endpoint of Let’s say the line segment AB is to be divided into 3 line segment project an arc to serve as an equal equal parts. Draw a line from A at any convenient distance from the endpoint. Create a bisector angle. Along this line use the compass to mark off 3 passing through the endpoint of the line segment arbitrary but equal lengths such that AC = CD = DE. and it will serve as the perpendicular line. Draw line EB. From D construct a line parallel to EB to intersect AB at G (perfectly feasible using straightedge and compass). Similarly, from C construct a line parallel to EB to intersect AB at F. Thus AB is divided into 3 equal sections AF, FG and GB. b. Dropping of perpendicular line. 4. Parallel lines - lines that never intersect even To drop a perpendicular from point P to line AB, set prolonged to a distance. Two lines having equal the needlepoint of the compass at P and strike an distances with each other. arc intersecting AB at arc 1. With the intersections a. Given line from a given distance of arc 1 and line AB as centers and any radius larger than one-half of assumed point of perpendicular intersection, strike arcs 2 & 3 to intersecting each b. From a given point 1 other. A line projected from P through the intersection is perpendicular to AB. c. From a given point 2 5. Triangles - a geometrical plane having three b. Erecting a Scalene Triangle. From a given sides and three angles. length of each side open your compass and a. Erecting an Isosceles Triangle. From a given intersect all of the three sides base project a compass where the opening equals that the required length of equal sides of a triangle. c. Erecting a Right Triangle. Same procedure will d. Erecting an Equilateral Triangle. From a given be applied by erecting perpendicular line to form a length of the base open the same on your compass 90 degree angle. and intersect to form a triangle or project a 60 degree triangle from the endpoints of the base. e. Inscribing an Equilateral Triangle. From a f. Circumscribing an Equilateral Triangle. Project given circle, project a vertical line passing through a 30 degree axis from the center of the circle and the center and intersecting above the the circumference to serve as tangent points of the circumference. Then project a 60 degree angle 60 degree angles. Then project a horizontal line from the intersection on both sides and connect the from the intersection of the axis and the intersection of the line and circumference below. circumference. 6. Squares - a quadrilateral having equal sides b. Inscribing a Square. Inscribe a square by and a 90 degree angle of its corners. connecting the intersections of the axis and the a. Erecting a Square. Construct a perpendicular circumference of the given circle (across corners). line on one side of the base and project an arc with Or project a 45 degree lines to intersect to the a measurement of the base to the erected circumference and connect the intersections. perpendicular line and do the same for the last corner of the square. c. Circumscribing a Square. Project a 7. Regular Polygons - a geometrical plane/s perpendicular line from the tangent points on the having equals sides and perpendicular to the circumference. Project an arc from the axis of the center of a circle. circle to the center and connect the intersections to a. Erecting a pentagon. A five-sided polygon form a square. Project a 45 degree axis and project projected from a given side by bisecting and perpendicular line from the tangent points on the projecting to locate the center of the circle as circumference. guide for the pentagon. b. Inscribing a Pentagon. Draw a horizontal axis AB c. Erecting and Inscribing a Hexagon. and a vertical axis CD. Locate bisector(by projecting arc 1 & 2) of the radius OB. Set a compass to the spread between bisector and point C, and, strike the arc 4. Set compass to the spread between C and intersection of arc 4 and line AO, and, with as a center, strike the arc 5 to the circumference. A line from C and the intersection on the circumference forms one side of the pentagon. Set the compass to this opening and lay off this interval from C around the circumference of the circle. Connect the points of the intersections on the circumference to form a pentagon. d. Circumscribing a Hexagon. e. Erecting a Heptagon. From a given circle, serve lie AB as radius. Create a semi-circle where point A as center as shown. Divide semi-circle into seven equal parts. Project rays from point A to intersections 1,2,3,4 and 5. Using line AB as radius, intersect an arc from B to line 8, 14-9, 15-10, 5-11, & 12-10. Connect the points as shown. f. Inscribing an Octagon. g. Circumscribing an Octagon. Thank You for Listening!