Appendix. Geometric Definitions Descriptive Geometry Principles Measuring Point System

Transcription

1 Geometric Definitions Descriptive Geometry Principles Measuring Point System A3 A6 A9 Architectural Drawing: A Visual Compendium of Types and Methods, 3rd Edition published by John Wiley & Sons, Inc by Rendow Yee. All rights reserved

2 Appendix Photo: Sundial Pedestrian Bridge Redding, California Architect: Santiago Calatrava Photo Courtesy of Charles Yim Photo: Oriental Pearl Tower Shanghai, China Architect: Jia Huan Cheng and the Shanghai Modern Architectural Design Co. Ltd. Photo Courtesy of Chris Jules Photo: Bellevue Art Museum Bellevue, Washington Architect: Steven Holl Photo Courtesy of Charles Yim All structures built, whether they are buildings or bridges, are geometric in nature. An understanding of basic shapes and their geometric properties enhances our appreciation of the drawing systems used in architecture such as descriptive geometry, paralines, and linear perspective. Euclid of Alexandria (b. 325 BC) was the mathematician of antiquity recognized as the father of modern geometry. Euclidean spaces are characterized by parallel lines that do not intersect. Non-Euclidean spaces are characterized by hyperbolic and elliptic geometry such as seen in the architecture of Calatrava, Coop Himmelb(l)au, Gehry and Hadid. Architectural Drawing: A Visual Compendium of Types and Methods, 3rd Edition published by John Wiley & Sons, Inc by Rendow Yee. All rights reserved. A2

3 Appendix Rendow Yee Frank Gehry s conceptual sketches always show a strong resemblance to his completed projects and are strong precursors to his final design concept. Drawing: Conceptual sketch by Frank O. Gehry Walt Disney Concert Center Los Angeles, California Courtesy of Frank O. Gehry Architect Architectural Drawing: A Visual Compendium of Types and Methods, 3rd Edition published by John Wiley & Sons, Inc by Rendow Yee. All rights reserved. A3

4 GEOMETRIC DEFINITIONS: TYPES OF ANGLES; PERPENDICULAR AND PARALLEL LINES; DETERMINATION OF A PLANE SURFACE; TYPES OF TRIANGLES A4

5 GEOMTERIC DEFINITIONS: PLANE SHAPES AND SOLID FORMS A5

6 GEOMETRIC DEFINITIONS: SOLID FORMS A6

7 WAREHOUSE ROOF: TRUE SHAPE AND COST STEP-BY-STEP SEQUENCE STEP (1): Find a true length view of the given ridge, B H D H. A line parallel to an image plane or folding-line will appear true length on that plane. Project corresponding point distances from the alternate adjacent plane (view). The warehouse shown above needs new roofing material. The owner wants you to estimate the cost for replacing the asphalt shingles on the roof. For this new building, asphalt shingles cost $2.50 per square foot. How much will it cost to lay shingles? Note: Always label all points to help you visualize the roof structure. Scale = 1/16 = 1-0 A7

8 WAREHOUSE ROOF: TRUE SHAPE AND COST STEP-BY-STEP SEQUENCE STEP (1): Find a true length view of the given ridge, B H D H. STEP (2): Find a point view of this true length line, D 2 B 2. A point view of a line must first appear as a true length in the adjacent view. When the lines of sight are parallel to the line, it will appear as a point. Any planes forming the true length line intersection will appear as an edge. Project corresponding point distances from the alternate adjacent plane (view). A8

9 WAREHOUSE ROOF: TRUE SHAPE AND COST STEP-BY-STEP SEQUENCE STEP (1): Find a true length view of the given ridge, B H D H. STEP (2): Find a point view of this true length line, D 2 B 2. This will give an edge view of both roof planes. STEP (3): Find the true shape of both roof planes, B 3 C 3 D 3 and B 3 A 3 D 3 STEP (4): Measure the roof areas and calculate the cost. COST CALCULATIONS AREA B 3 C 3 D 3 = 294 SQ. FT. AREA B 3 A 3 D 3 = 560 SQ. FT. 294 X $2.50 = $ X $2.50 = $ $ To obtain true shapes, construct folding lines parallel to the edge views of the planes. True sizes occur when the lines of sight are perpendicular to the plane. Project corresponding points as in Steps 1 and 2. A9

10 Note: To better understand pages in the book on measuring points and establishing scale in two-point perspectives, read pages A10-A16 on vertical vanishing lines, diagonal vanishing points, determining measuring points, and completing a cube using the measuring point system. Diagram and text: Courtesy of William R. Benedict, Assistant Professor California Polytechnic State University College of Architecture & Environmental Design San Luis Obispo, California Vertical Vanishing Lines Vertical vanishing lines (VVL) must be drawn through both vanishing points (VP). These are the vanishing lines for all planes and lines parallel to the vertical faces of the cube. We now have the information necessary to locate the 45 diagonal vanishing points for the two vertical surfaces of the cube or any surfaces parallel to them. The 45 diagonal vanishing points are the points to which the two sets of 45 lines lying on a particular plane will vanish. They are located on the vanishing line for the plane. A10

11 Diagram and text: Courtesy of William R. Benedict, Assistant Professor California Polytechnic State University College of Architecture & Environmental Design San Luis Obispo, California Constructing Diagonal Vanishing Points Transfer the distance from each VP to the SP to the vertical vanishing line passing through the respective VP. This can be done with a compass, as shown in the illustration, or with a scale. The 45 diagonal vanishing points can be generated for each plane one will be above and one below their respective vanishing points. One diagonal vanishing point for each plane is often all that is necessary for construction, as indicated in the illustration. A11

12 Diagram and text: Courtesy of William R. Benedict, Assistant Professor California Polytechnic State University College of Architecture & Environmental Design San Luis Obispo, California Constructing Cube Faces Draw lines from the DVPs to one end of the known vertical edge of the cube. Use the DVP that is above or below the VP for horizontal lines on the same plane. The diagonal lines are 45 lines; therefore, their intersection with a vanishing edge transfers the length of the vertical edge to the horizontal that is, it defines a square on each surface. Draw vertical lines through the intersections to establish the vertical faces of the cube. A12

13 Top drawing: Horyuji Museum, Ueno Park, Tokyo, Japan Design Office: Tanaguchi Associates, Tokyo, Japan 20 x 30 (50.8 x 76.2 cm), Medium: AutoCAD, 20 x 30 (50.8 x 76.2 cm) Courtesy of Lawrence Ko Leong, Architectural Illustrator Cube Completion Vanish lines to define the top face of the cube and add line weight and value to complete the cube. Additional cubes can be generated using the diagonal. A13

14 Diagram and text: Courtesy of William R. Benedict, Assistant Professor California Polytechnic State University College of Architecture & Environmental Design San Luis Obispo, California Determining Measuring Points Cubes as well as other geometric forms can be constructed simply by using measuring points (MP). Measuring points are the points used to transfer scaled dimensions from the horizontal measuring line to lines vanishing through the intersection of the horizontal and vertical measuring lines. Proportional sizes can be transferred with the measuring points from any horizontal line to vanished lines passing through the intersection of that horizontal line and a vertical line. Measuring points offer an additional way of introducing dimensions into a perspective. The process begins with the construction of the horizon line, ground line, vanishing points, mid-point, the perspective field of view, and so forth. Transfer the distances from VPR to SP and VPL to SP down to the horizon line. This can be done with a compass, as shown in the illustration, or with a scale. The points on the horizon line resulting from this transfer are the measuring points for the perspective setup. In two-point perspective, a shown above, there are two measuring points; whereas in one-point perspective there is only one measuring point, which is also called a diagonal vanishing point. A14

15 Diagram and text: Courtesy of William R. Benedict, Assistant Professor California Polytechnic State University College of Architecture & Environmental Design San Luis Obispo, California Entering Dimensions Enter the horizontal dimensions that you want to transfer into the perspective. The dimensions are entered on a horizontal measuring line (HML). In the example, an arc is used to transfer the height of the cube to either side of its existing lead edge and vertical measuring line. A horizontal measuring line (HML) may be located anywhere along any vertical line within the perspective. The only requirements are (1) that the vertical line can be used to establish the measurement scale or is a vertical measuring line (VML), and (2) that a pair of lines parallel to the X and Y axes be drawn through the intersection of the vertical line and the horizontal measuring line. Constructing Cube Faces The measuring points are now used to transfer the dimensions from the horizontal measuring line to the X and Y axes or lines parallel to them. The intersection of the line drawn from the dimension on the HML to the appropriate MP with the axis transfers the dimension into the perspective. Draw a vertical line through the intersections to define the two faces. Note: When transferring a dimension back into the perspective when making it smaller you use the MP on the opposite side of the vertical line. When transferring it forward making it bigger you use the MP on the same side. A15

16 Diagram and text: Courtesy of William R. Benedict, Assistant Professor California Polytechnic State University College of Architecture & Environmental Design San Luis Obispo, California Cube Completion Vanish lines to define the top surface of the cube, and add line weight and value to complete the cube. This approach brings together the two halves of the measuring point system the diagonal vanishing points and the measuring points. Once the initial volume is established, any combination of vanishing points and dividing, multiplying and transferring techniques can be employed along with the measuring points to develop the perspective. The goal is to make choices that complete the drawing as efficiently as possible. A16

17 Diagram and text: Courtesy of William R. Benedict, Assistant Professor California Polytechnic State University College of Architecture & Environmental Design San Luis Obispo, California This illustration brings together the basic elements of the measuring point system. Notice that the arc defining the diagonal vanishing points is the same one that defines the measuring points. The complexity of the drawing makes a good case for introducing the concepts in a systematic and gradual manner to avoid being caught in the web of lines and concepts. The link between the diagonal vanishing points and the measuring points is that they are both established with the same dimension. You can employ the diagonal vanishing points, measuring points, and multiplication, division and transfer techniques in any perspective construction. The choice depends on the strategy that will produce the desired results with the greatest efficiency. A17