Lecture 4: Engineering Geometry

İİTTÜÜ--SUNY SUNY 20062006--20072007 FallFall

Graphic Graphic CommunicationsCommunications

Assoc. Assoc. Prof.DrProf.Dr. Cengizhan . Cengizhan İİpbpbüükerker&&

Modified by Assist. Prof. Dr. M. Zeki COModified by Assist. Prof. Dr. M. Zeki COŞŞKUNKUN

Engineering Geometry

Engineering geometry is the basic geometric elements and forms used in engineering design such as lines, circles and planes.

Engineering and technical graphics are concernedwith the descriptions of shape, size, and operationof engineered products.

The shape description of an object relates to the positions of its component geometric elements in space.

Shape description

Shape description of an object relatesthe positions of its component geometricelements (e.g., vertices, edges, faces) in space.

CoordinateCoordinate SpaceSpaceIn order to locate points, lines, planes, or other geometric forms, their positions must first be referenced to some known position, called a reference point or origin of measurement.

CartesianCartesianCoordinateCoordinate SystemSystem

The Cartesian coordinate system, commonly used in mathematics and graphics, locates the positions of geometric forms in 2-D and 3-D space.

TheThe rightright handhand rulerule

The The rightright--hand rulehand rule is used to determine is used to determine the positive direction of the axes. The rightthe positive direction of the axes. The right--hand rule defines the X, Y, and Z axes, as hand rule defines the X, Y, and Z axes, as well as the positive and negative directions well as the positive and negative directions of rotation on each axes.of rotation on each axes.

CoordinateCoordinateSystemsSystems

••Polar coordinatesPolar coordinates are used to locate points in the Xare used to locate points in the X--Y Y plane. Polar coordinates specify a distance and an angle plane. Polar coordinates specify a distance and an angle from the origin (0,0).from the origin (0,0).••Cylindrical coordinatesCylindrical coordinates locate a point on the surface locate a point on the surface of a cylinder by specifying a distance and an angle in the of a cylinder by specifying a distance and an angle in the XX--Y plane, and the distance in the Z direction.Y plane, and the distance in the Z direction.••Spherical coordinatesSpherical coordinates locate a point on the surface locate a point on the surface of a sphere by specifying an angle in one plane, an of a sphere by specifying an angle in one plane, an angle in another plane, and one height.angle in another plane, and one height.

CoordinateCoordinate SystemSystem••Absolute coordinatesAbsolute coordinates are always referenced to are always referenced to the origin (0,0,0).the origin (0,0,0).••Relative coordinatesRelative coordinates are always referenced to a are always referenced to a previously defined location. previously defined location. ••The The world coordinate systemworld coordinate system uses a set of three uses a set of three numbers (x,y,z) located on three mutually numbers (x,y,z) located on three mutually perpendicular axes and measured from the origin perpendicular axes and measured from the origin (0,0,0). (0,0,0). ••The The local coordinate systemlocal coordinate system is a moving system is a moving system that can be positioned anywhere in 3that can be positioned anywhere in 3--D space by the D space by the user, to assist in the construction of geometry.user, to assist in the construction of geometry.

GeometricGeometricElementsElements

Geometric elements are categorized as: Geometric elements are categorized as: points, lines, surfacespoints, lines, surfaces, or , or solidssolids. Lines, . Lines, surfaces, and solids also have many surfaces, and solids also have many subcategories.subcategories.

PointPoint

A A pointpoint is a theoretical location that has neither is a theoretical location that has neither width, height, nor depth. Points describe an exact width, height, nor depth. Points describe an exact location in space. Normally, a point is represented location in space. Normally, a point is represented in technical drawings as a small cross made of in technical drawings as a small cross made of dashes that are approximately 1/8" long. In dashes that are approximately 1/8" long. In computer graphics, it is common to use the word computer graphics, it is common to use the word nodenode to mean a point. A to mean a point. A locuslocus represents all represents all possible positions of a point.possible positions of a point.

LineLine

A A lineline is a geometric primitive that has length is a geometric primitive that has length and direction, but not thickness. A line may be and direction, but not thickness. A line may be straight, curved, or a combination of these. straight, curved, or a combination of these.

A A straight linestraight line is generated by a point moving is generated by a point moving in a constant direction. in a constant direction.

A A straight finite linestraight finite line is a line of specific length. is a line of specific length.

A A straight infinite linestraight infinite line is a line of nonspecific is a line of nonspecific length. length.

LineLine typestypes

••Parallel lineParallel line

••NonNon--parallel lineparallel line

••Perpendicular linePerpendicular line

••Tangent lineTangent line

••Intersecting lineIntersecting line

••Curved lineCurved line

CurvedCurved lineline

A A curved linecurved line is the path generated by a point moving in a is the path generated by a point moving in a constantly changing direction, or is the line of intersection constantly changing direction, or is the line of intersection between a 3between a 3--D curved surface and a plane. D curved surface and a plane.

On a On a singlesingle--curved linecurved line, all points of the line are in a plane. , all points of the line are in a plane.

On a On a doubledouble--curved linecurved line, no four consecutive points are in , no four consecutive points are in the same plane. the same plane.

A A regular curveregular curve is a constantis a constant--radius arc or circle generated radius arc or circle generated around a single center point.around a single center point.

A A circlecircle is a singleis a single--curvedcurved--surface primitive, all surface primitive, all points of which are equidistant from one point, points of which are equidistant from one point, the center. A circle is also created when a plane the center. A circle is also created when a plane passes through a right circular cone or cylinder passes through a right circular cone or cylinder and is perpendicular to the axis of the cone.and is perpendicular to the axis of the cone.

CircleCircle

ElementsElements of a of a circlecircle

••Center.Center.••Circumference. Circumference. ••Radius. Radius. ••Diameter. Diameter. ••Sector.Sector.••Semicircle. Semicircle. ••Quadrant. Quadrant. ••Sector. Sector. ••Segment.Segment.

ElementsElements related with arelated with a circlecircle

••Chord. Chord. ••Semicircle. Semicircle. ••Minor arc. Minor arc. ••Major arc.Major arc.••Segment. Segment. ••Tangent. Tangent. ••Concentric circles. Concentric circles. ••Eccentric circles. Eccentric circles. ••CircumscribedCircumscribed

ConicConic curvescurves

Conic curvesConic curves, or , or conicsconics, are special case , are special case singlesingle--curved lines that can be described in curved lines that can be described in several ways: as sections of a cone; as several ways: as sections of a cone; as algebraic equations; and as the loci of points. algebraic equations; and as the loci of points. For our purposes, conics are the curves formed For our purposes, conics are the curves formed by the intersection of a plane with a right by the intersection of a plane with a right circular cone and include the ellipse, parabola, circular cone and include the ellipse, parabola, and hyperbola.and hyperbola.

EllipseEllipse

Mathematically, an Mathematically, an ellipseellipse is the set of all points in a plane is the set of all points in a plane for which the sum of the distances from two fixed points for which the sum of the distances from two fixed points (the foci) in the plane is constant. (the foci) in the plane is constant.

The The major diametermajor diameter (major axis) of an ellipse is the (major axis) of an ellipse is the longest straightlongest straight--line distance between the sides and is line distance between the sides and is through both foci.through both foci.

The The minor diameterminor diameter (minor axis) is the shortest straight(minor axis) is the shortest straight--line distance between the sides and is through the bisector line distance between the sides and is through the bisector of the major axis. of the major axis.

The The foci foci are the two points used to construct the perimeter are the two points used to construct the perimeter and are on the major axis.and are on the major axis.

RoulettesRoulettes

RoulettesRoulettes are the curves generated by the rolling contact are the curves generated by the rolling contact of one curve or line on another curve or line. Any point of one curve or line on another curve or line. Any point attached to the rolling curve or line will describe the attached to the rolling curve or line will describe the roulette curve. The moving point is called the generating roulette curve. The moving point is called the generating point.point.

A A spiralspiral is a singleis a single--curved surface that begins at a point curved surface that begins at a point called a pole and becomes larger as it travels around the called a pole and becomes larger as it travels around the origin in a plane. origin in a plane.

A A cycloidcycloid is the curve generated by the motion of a point is the curve generated by the motion of a point on the circumference of a circle as the circle is rolled along on the circumference of a circle as the circle is rolled along a straight line in a plane. a straight line in a plane.

FreeformFreeform curvescurves

Simple curves are circles, arcs, and ellipses. More complex Simple curves are circles, arcs, and ellipses. More complex curves used in engineering design are called freeform curves used in engineering design are called freeform curves.curves.A A spline curvespline curve is a smooth, freeform curve that connects is a smooth, freeform curve that connects a series of control points. Changing any single control a series of control points. Changing any single control point will result in a change in the curve, so that the curve point will result in a change in the curve, so that the curve can pass through the new point. The can pass through the new point. The Bezier curveBezier curve, which , which uses a set of control points that only approximate the uses a set of control points that only approximate the curve. The curve. The BB--spline curvespline curve, which approximates a curve , which approximates a curve to a set of control points and does provide for local to a set of control points and does provide for local control.control.

AnglesAngles

AnglesAngles are formed by the apex of two intersecting lines or are formed by the apex of two intersecting lines or planes. Angles are categorized by their degree planes. Angles are categorized by their degree measurement.measurement.

••Straight angleStraight angle. An angle of 180 degrees. . An angle of 180 degrees. ••Right angleRight angle. An angle of 90 degrees. . An angle of 90 degrees. ••Acute angleAcute angle. An angle of less than 90 degrees.. An angle of less than 90 degrees.••Obtuse angleObtuse angle. An angle of more than 90 . An angle of more than 90 degrees.degrees.••Complementary anglesComplementary angles. Two adjacent angles . Two adjacent angles whose sum equals 90 degrees. whose sum equals 90 degrees.

PlanesPlanes

A A planeplane is an infinite, unbounded, twois an infinite, unbounded, two--dimensional surface dimensional surface that wholly contains every straight line joining any two that wholly contains every straight line joining any two points lying on the surface. A plane can be defined by:points lying on the surface. A plane can be defined by:

••three points not in a straight linethree points not in a straight line••two parallel linestwo parallel lines••a line plus a point that is not on the line or its extensiona line plus a point that is not on the line or its extension••two intersecting linestwo intersecting lines

SurfaceSurfaceA A surfacesurface is a finite portion of a plane, or the outer face of is a finite portion of a plane, or the outer face of an object bounded by an identifiable perimeter. Surfaces are an object bounded by an identifiable perimeter. Surfaces are generally classed as: generally classed as:

••planar planar ••singlesingle--curved curved ••doubledouble--curved curved ••warped warped ••freeform. freeform.

Surfaces can also be classified asSurfaces can also be classified as••ruled ruled ••developable or undevelopabledevelopable or undevelopable

22--D D SurfacesSurfaces

22--D surfaces D surfaces can be classified by the characteristics of can be classified by the characteristics of their sides.their sides.

••QuadrilateralsQuadrilaterals••Polygons Polygons ••TrianglesTriangles

QuadrilateralsQuadrilateralsQuadrilateralsQuadrilaterals are fourare four--sided plane figures of any shape. sided plane figures of any shape. The sum of the angles inside a quadrilateral will always The sum of the angles inside a quadrilateral will always equal 360 degrees. Quadrilaterals are classified by the equal 360 degrees. Quadrilaterals are classified by the characteristics of their sides.characteristics of their sides.

••SquareSquare. . ••RectangleRectangle. . ••RhombusRhombus. . ••RhomboidRhomboid. . ••Regular trapezoidRegular trapezoid. . ••Irregular trapezoidIrregular trapezoid. . ••TrapeziumTrapezium. .

PolygonsPolygons

A A polygonpolygon is a multisided plane of any number of is a multisided plane of any number of sides. If the sides of the polygon are equal in length, sides. If the sides of the polygon are equal in length, the polygon is called a the polygon is called a regular polygonregular polygon. Regular . Regular polygons are grouped by the number of sides.polygons are grouped by the number of sides.

TriangleTriangle

••A A triangletriangle is a polygon with three sides.is a polygon with three sides. Triangles are Triangles are named according to their angles or the number of equal named according to their angles or the number of equal sides. sides.

••Equilateral triangleEquilateral triangle. .

••Isosceles triangleIsosceles triangle. .

••Scalene triangleScalene triangle. .

••Right triangleRight triangle. .

••Obtuse triangleObtuse triangle. .

••Acute triangleAcute triangle. .

RuledRuled surfacessurfacesPolyhedra, singlePolyhedra, single--curved curved surfaces, and warped surfaces, and warped surfaces are classified as surfaces are classified as ruled surfaces. All ruled ruled surfaces. All ruled surfaces, except for warped surfaces, except for warped surfaces, are developable. surfaces, are developable. The cone, cylinder, and The cone, cylinder, and convolute are the only types convolute are the only types of surfaces in this class and of surfaces in this class and all are developable.all are developable.

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PolyhedronPolyhedron

••A A polyhedronpolyhedron is a symmetrical or asymmetrical 3is a symmetrical or asymmetrical 3--D D geometric surface or solid object with multiple polygonal geometric surface or solid object with multiple polygonal sides. Regular polyhedra are classified by the shape and sides. Regular polyhedra are classified by the shape and number of faces, as follows:number of faces, as follows:

••TetrahedronTetrahedron..••HexahedronHexahedron. . ••OctahedronOctahedron. . ••DodecahedronDodecahedron....••IcosahedronIcosahedron. .

PyramidPyramid

A A pyramidpyramid is a polyhedron that has a polygon is a polyhedron that has a polygon for a base and lateral faces that have a for a base and lateral faces that have a common intersection point called a vertex. The common intersection point called a vertex. The axis of a pyramid is the straight line connecting axis of a pyramid is the straight line connecting the center of the base to the vertex.the center of the base to the vertex.••If the axis is perpendicular to the base, the If the axis is perpendicular to the base, the pyramid is a pyramid is a right pyramidright pyramid••If the axis is not perpendicular to the bases, If the axis is not perpendicular to the bases, the pyramid is an the pyramid is an oblique pyramidoblique pyramid..

UndevelopableUndevelopable surfacessurfaces

A A warped surfacewarped surface is a doubleis a double--curved ruled 3curved ruled 3--D D surface generated by a straight line moving such that surface generated by a straight line moving such that any two consecutive positions of the line are skewed any two consecutive positions of the line are skewed (not in the same plane). Warped surfaces are not (not in the same plane). Warped surfaces are not developable.developable.

FractalsFractals are an example of geometries which are an example of geometries which exhibit a degree of selfexhibit a degree of self--similarity but which are quite similarity but which are quite complex and random.complex and random.

ReferencesReferences

GaryGary R. R. BertolineBertoline & & EricEric N. N. WiebeWiebeFundamentalsFundamentals of of GraphicsGraphics CommunicationCommunication, , 3/e, 3/e, McGrawMcGraw--HillHill CollegeCollege, 2001, , 2001, ISBNISBN00723220980072322098

http://http://higheredhighered..mcgrawmcgraw--hillhill.com/.com/sitessites/0072322098//0072322098/