Instructor: Laura Gerold, PE Catalog #10614113 Class # 22784, 24113, 24136, & 24138Class Start: January 18, 2012 Class End: May 16, 2012
Project Proposals are due in one week on February 22nd Extra Credit is due in one week on February 22nd Project – Have one view (out of six total) drawn on non-grid paper Homework assignment is now on Blackboard in Class Materials/ Homework folder
Small errors during drawing construction can lead to big errors in your end product . . .
Land Surveying is "The art and science of measuring angles and distances on or near the surface of the earth.“
Originally built cabin in 1819, but had to build a new one in 1822 dueto surveying error.
Construction started on the fort in 1816 to protect against invasions of theBritish/Canadians, but it was soon discovered the fort was entirely inCanada and not in the US due to a surveying error.
Break into groups of 2 or 3 and refresh yourselves how to draw these three items, be prepared to explain how you did the work . . . Triangle Draw a triangle with two sides given, R = 2”, S = 1” Triangle Draw a triangle with sides 5”, 4.5”, and 4” Bisect the three interior angles The bisectors should meet at a point Draw a circle inscribed in the triangle with the point where the bisectors meet in the center Drawing a point through a line and perpendicular to a line Draw a line Draw a point on the line Draw a line through the point and perpendicular to the line Repeat process, but this time put the point not on the line
1. Lightly draw arc CR 2. Lightly draw equal arcs r with radius slightly larger than half BC, to intersect at D 3. Draw line AD, which bisects the angle
Drawing a right triangle with hypotenuse and one side given1. Given sides S and R2. With AB as diameter equal to S, draw a semicircle3. With A as center, R as radius, draw an arc intersecting the semicircle C.4. Draw AC and CB
Drawing a Triangle with Sides Given1. Draw one side, C2. Draw an arc with radius equal to A3. Lightly draw an arc with radius equal to B4. Draw sides A and B from the intersection of the arcs
When the Point is Not on the Line (AB & P given) From P, draw any convenient inclined line, PD on (a) Find center, C, of line PD Draw arc with radius CP, center at C Line EP is required perpendicular P as center, draw an arc to intersect AB at C and D (b) With C & D as centers and radius slightly greater than half CD, draw arcs to intersect at E Line PE is required perpendicular When the Point Is Not on the Line When the Point Is on the Line T-square Method
When the Point is on the Line (AB & P given) With P as center and any radius, strike arcs to intersect AB at D and G (c) With D and G as centers and radius slightly greater than half DG, draw equal arcs to intersect at F. Line PF is the required perpendicular When the Point Is Not on the Line When the Point Is on the Line T-square Method
For measuring or setting off angles other than those obtainable withtriangles, use a protractor. Plastic protractors are satisfactory for most angular measurementsNickel silver protractors areavailable when high accuracyis required
Math Made Easy: Measuring Angles (part 1) Math Made Easy: Measure Angles (part 2) Many angles can be laid out directly with the protractor.
Tangent Method1. Tangent = Opposite / Adjacent2. Tangent of angle is y/x3. Y = x tan4. Assume value for x, easy such as 105. Look up tangent of and multiply by x (10)6. Measure y = 10 tan
Sine Method1. Sine = opposite / hypotenuse2. Sine of angle is y/z3. Draw line x to easy length, 104. Find sine of angle , multiply by 105. Draw arc R = 10 sin
Chord Method1. Chord = Line with both endpoints on a circle2. Draw line x to easy length, 103. Draw an arc with convenient radius R4. C = 2 sin ( /2)5. Draw length C
Draw two lines forming an angle of 35.5 degrees using the tangent, sine, and chord methods Draw two lines forming an angle of 40 degrees using your protractor
Side AB given With A & B as centers and radius AB, lightly construct arcs to intersect at C Draw lines AC and BC to complete triangle
Draw a 2” line, AB Construct an equilateral triangle
1. One side AB, given2. Draw a line perpendicular through point A3. With A as center, AB as radius, draw an arc intersecting the perpendicular line at C4. With B and C as centers and AB as radius, lightly construct arcs to intersect at D5. Draw lines CD and BD
Diameters Method1. Given Circle2. Draw diameters at right angles to each other3. Intersections of diameters with circle are vertices of square4. Draw lines
Lightly draw a 2.2” diameter circle Inscribe a square inside the circle Circumscribe a square around the circle
Geometric Method 1. Bisect radius OD at C 2. Use C as the center and CA as the radius to lightly draw arc AE 3. With A as center and AE as radius draw arc EB 4. Draw line AB, then measure off distances AB around the circumference of the circle, and draw the sides of the Pentagon through these points Dividers Method Geometric Method
Lightly draw a 5” diameter circle Find the vertices of an inscribed regular pentagon Join vertices to form a five-pointed star
Each side of a hexagon is equal to the radius of the circumscribed circle Use a compass Centerline Variation Steps
Method 1 – Use a Compass Each side of a hexagon is equal to the radius of the circumscribed circle Use the radius of the circle to mark the six sides of the hexagon around the circle Connect the points with straight lines Check that the opposite sides are parallel Use a compass
Method 2 – Centerline Variation Draw vertical and horizontal centerlines With A & B as centers and radius equal to that of the circle, draw arcs to intersect the circle at C, D, E, and F Complete the hexagon Centerline Variation
Lightly draw a 5” diameter circle Inscribe a hexagon
Given a circumscribed square, (the distance “across flats”) draw the diagonals of the square. Use the corners of the square as centers and half the diagonal as the radius to draw arcs cutting the sides Use a straight edge to draw the eight sides
Lightly draw a 5” diameter circle Inscribe an Octagon
A,B, C are given points not on a straight line Draw lines AB and BC (chords of the circle) Draw perpendicular bisectors EO and DO intersecting at O With center at ), draw circle through the points
Draw three points spaced apart randomly Create a circle through the three points
Method 1 This method uses the principle that any right triangle inscribed in a circle cuts off a semicircle Draw any cord AB, preferably horizontal Draw perpendiculars from A and B, cutting the circle at D and E Draw diagonals DB and EA whose intersection C will be the center of the circle
Method 2 – Reverse the procedure (longer) Draw two nonparallel chords Draw perpendicular bisectors. The intersection of the bisectors will be the center of the circle.
Draw a circle with a random radius on its own piece of paper Give your circle to your neighbor Find the center of the circle given to you
Given a line AB and a point P on the line At P, draw a perpendicular to the line Mark the radius of the required circle on the perpendicular Draw a circle with radius CP
Draw a 4” long line Place a point P at the midpoint of the line Draw a 2” diameter circle tangent to the line at P
Work as a 2-3 person group to figure out the following problems without a T-square Given a point on a circle, draw a line tangent to the circle Given a point not on the circle, draw a line tangent to the circle and through the point
Method 1(a) Given line AB, point P, radius R Draw line DE parallel to given line and distance R from it From P draw arc with radius R, cutting line DE at C C is the center of the required tangent arc Tangents
Method 2 (b) Given line AB, tangent point Q on the line, and point P Draw PQ, the chord of the required arc Draw perpendicular bisector BE At Q, draw a perpendicular to the line to intersect DE at C C is the center of the required tangent arc Tangents
Method 3 (c) Given arc with center Q, point P, and radius R From P, strike an arc with radius R From Q draw an arc with radius equal to given arc plus R The intersection C of the arcs is the center of tangent arc Tangents
Given line AB= 3” long, any point P not the line (similar to graphic a on page 4.28), and radius 2”, draw an arc tangent to AB through point P.
Two lines are given at right angles to each other With given radius, R, draw an arc intersecting the given lines at tangent points T With given radius R again, and with points T as centers, draw arcs intersecting at C With C as center and given radius R, draw the required tangent arc For small radii, such as 1/8R for fillets and rounds, it is not practicable to draw complete tangency constructions. Instead, draw a 45° bisector of the angle and locate the center of the arc by trial along this line
Draw two intersecting at right angles vertical and horizontal lines, each 2.5 inches long Draw a 1.5 inch radius arc tangent to the two lines
Given two lines not making a 90° Draw lines parallel to the given lines at distance R from them to intersect at C the center From C, drop perpendiculars to the given lines to locate tangent points, T With C as the center and with given radius R, draw the required tangent arc between the points of tangency
Draw two intersecting vertical and horizontal lines as an acute angle, each 2.5 inches long Draw a 1.5 inch radius arc tangent to the two lines
Given arc with radius G and a straight line AB Draw a straight line and an arc parallel to the given straight line at the required radius distance R from them. Will intersect at C, required center From C, draw a perpendicular to the given straight line to find one point of tangency, T. Join the centers C and O with a straight line to locate the other point of tangency With the center C and radius R, draw the required tangency arc between the points of tangency
Given arcs with centers A and B and required radius R With A and B as centers, draw arcs parallel to the given arcs and at a distance R from them; their intersection C is the center of the required tangent arc Draw lines of the centers AC and BC to locate points of Tangency, T, and draw the required tangent arc between the points of tangency
Required Arc to Enclose Two Given Arcs With A & B as centers lightly draw arcs HK-r (given radius minus radius of small circle) and HK-R (given radius minus radius of large circle) intersecting at G, the center of the required tangent arc Lines of centers GA and GB (extended) determine points of tangency, T
Required Arc to Enclose One Given Arc With C & D as centers, lightly draw arcs HK+r (given radius plus radius of small circle) and HK-R (given radius minus radius of large circle) intersecting at G, the center of the required tangent arc Lines extended through centers GC and GD determine the points of tangency, T
Group project - Work together to do the following: Draw an arc tangent to an arc and a straight line using your own measurements Draw an arc tangent to two arcs using your own measurements Draw an arc tangent to two arcs and enclosing one or both
Ogee is a curve shaped somewhat like an S, consisting of two arcs that curve in opposite senses, so that the ends are parallel. Used in: Molding Architecture Marine Timber Construction
Connecting Two Parallel Lines (Method 1) Let NA and BM be 2 parallel lines Draw AB and assume inflection point T (midpoint) At A and B, draw perpendiculars AF and BC Draw perpendicular bisectors at AT and BT Intersections F and C of the bisectors and the perpendiculars are the centers of the tangent circles Connecting Two Parallel Lines Connecting Two Nonparallel Lines
Connecting Two Parallel Lines (Method 2) Let AB and CD be two parallel lines with point B as one end of curve and R the given radii At B, draw the perpendicular to AB, make BG = R, and draw the arc as shown Draw SP parallel to CD at distance R from CD With center G, draw the arc of the radius 2R, interesting line SP and O. Draw perpendicular OJ to locate tangent point J, and join centers G & O to locate point of tangency T. Use centers G & O and radius R to draw tangent arcs Connecting Two Parallel Lines Connecting Two Nonparallel Lines
Connecting Two Nonparallel Lines Let AB and CD be two nonparallel lines Draw the perpendicular to AB at B Select point G on the perpendicular so that BG equals any desired radius, and draw the arc as shown (c) Draw the perpendicular CD at C and make CE=BG Join G to E and bisect it. Intersection F of the bisector and the perpendicular CD, extended, is the center of the second arc. Join the centers of the two arcs to locate tangent point T, the inflection of the curve Connecting Two Parallel Lines Connecting Two Nonparallel Lines
Draw two parallel lines Draw an ogee curve using method 1 or 2
The conic sections are curves produced by planes intersecting a right circularcone. Four types of curves are produced: the circle, ellipse, parabola, and hyperbola, according to the position of the planes.
The curves are largely successive segments of geometriccurves, such as the ellipse, parabola, hyperbola, and involute.
These ellipse guides are usually designated by the ellipse angle, theangle at which a circle is viewed to appear as an ellipse.
Major axis = long axis of ellipse Minor axis = short axis of ellipse The foci of the ellipse are two special points E and F on the ellipses major axis and are equidistant from the center point. The sum of the distances from any point P on the ellipse to those two foci is constant and equal to the major axis ( PE + PF2= 2A ). Each of these two points is called a focus of the ellipse.
Let AB be the major axis and CD the minor axis To find foci E and F, draw arcs R with radius equal to half the major axis and centers at the end of the minor axis Between E and O on the major axis, mark at random a number of points. Using a random point (point 3), with E and F as centers and radii A-3 and B-3, draw arcs to intersect at four points 3’. Use the remaining points to find four additional points on the ellipse in the same manner. Sketch the ellipse lightly through the points
For many purposes, particularly where a small ellipse is required, use the approximate circular arc method. Given axes AB & CD Draw line AC. With O as center and OA as radius, draw arc AE. With C as center and CE as radius, draw arc EF. Draw perpendicular bisector GH of the line AF; the points K & J, where they intersect the axes are centers of the required arc Find centers M & L by setting off OL = OK and OM = OJ. Using centers K, L, M, & J, draw circular arcs as shown. The points of tangency T are at the junctures of the arcs n the lines joining the centers.
Draw a major axis 5” long and a minor axis 2.5” long. Draw an ellipse by the foci method with at least five points in each quadrant Use the same axes and use the approximate ellipse method
The curve of intersection between a right circular cone and a plane parallelto one of its elements is a parabola.
Given focus F and directrix AB Draw a line DE parallel to the directrix and at any distance CZ from it With center at F and radius CA, draw arcs to intersect line DE at the points Q & R (points on the parabola) Proceed in the same manner to determine as many points as needed
Draw a parabola with a vertical axis and focus 0.5” from the directrix. Find at least 9 points on the curve
Look at Exercise 4.60 and talk through methods that you would use to create this drawing in pencil Present these methods to class
To make and interpret drawings you need to know how to create projections and understand the standard arrangement of views.You also need to be familiar with the geometry of solid objects and be ableto visualize a 3D object that is represented in a 2D sketch or drawing.
Vanishing Points: An Introduction to Architectural Drawing
The system of views is calledmultiview projection. Each viewprovides certain definiteinformation. For example, a frontview shows the true shape andsize of surfaces that are parallelto the front of the object.
The system of views is called multiview projection. Each view providescertain definite information.
Any object can be viewed from six mutually perpendiculardirections,
Revolving the Object to Produce Views. You can experiencedifferent views by revolving an object.
The three principal dimensions of an object are width, height, and depth. The front view shows only the height and width of the object and not the depth. In fact, any principal view of a 3D object shows only two of the three principal dimensions; the third is found in an adjacent view. Height is shown in the rear, left-side, front, and right-side views. Width is shown in the rear, top, front, and bottom views. Depth is shown in the left-side, top, right-side, and bottom views.
The outline on the plane of projection shows how the object appears to the observer.In orthographic projection, rays (or projectors) from all points on the edges or contoursof the object extend parallel to each other and perpendicular to the plane of projection.The word orthographic means “at right angles.” Projection of an Object
Specific names are given to the planes of projection. The front view isprojected to the frontal plane. The top view is projected to the horizontalplane. The side view is projected to the profile plane.
• Chapter 5 – Orthographic Projection• Project Proposal due next week (February 22nd)
On one of your sketches, answer the following two questions: What was the most useful thing that you learned today? What do you still have questions about?
Chapter 4 Exercises (note they are in mm) 4.11,4.13, 4.15, 4.18, 4.20, 4.22, 4.30, 4.33, 4.35 (major axis should be 100 mm not 10 mm), 4.40, 4.58