DDF M1 Hyperbolic Paraboloid

DIGITAL DESIGN + FABRICATION SM1, 2016 M1 JOURNAL - HYPERBOLIC PARABOLOID CHO HIN TING FRANKIE 804015 TUTOR: TIM CAMERON

1

Measured Drawings m

180m

MEASUREMENT METHODOLOGY

200mm

The hyperbolic paraboloid origami was folded using 20x20cm 110gsm paper with a pattern of two diagonal folds and 14 horizontal and vertical folds. It was then photographed using a Nikon 135mm zoom lens to minimize perspective in the orthographic view, and the lengths were digitally measured and compared against ruler measurements. Mathematical calculations are utilized to crosscheck the values.

180

mm

PERSPECTIVE VIEW NOT ON SCALE 2

18 8

10

12

14

m .5m 12

16

ELEVATION SCALE: 1:2

0

14

16

2

4

scale in centimeters

17.6mm

6

35mm

scale in centimeters

PLAN SCALE: 1:2

0

2

4

6

8

10

12

18

20

3

Analysis θ = 90o

#1. A quarter of the paper before folding

l =10cm

w = 20cm

θ > 90o #2. A quarter of the paper after folding

l <10cm

(not on scale)

w = 20cm

4

Why is the paraboloid three-dimensional?

Simplified representation of the paraboloid

The unfolded square paper is able to maintain two-dimensional when the centre of the paper at 360o (i.e. each quarter at right angle). When the paper is folded zigzag, the distance between the center of the paper and the edge (denoted by l) reduces, while the length of the edge (denoted by w) remains unchanged. This means the center angle (4θ) increase in size, exceeding 360o. Due to this, the paper is unable to maintain within twodimensions.

The distinctive curvature of the hyperbolic paraboloid is formed by the chronologically changing angles of the straight lines in a 3D space. Here, the curvature of the paraboloid is modelled by connecting 8 straight lines along two perpendicularly placed edges, each connected on the same position on the opposite edge. The resolution of the curve increases as the density of lines increase. Although the paraboloid shows a curved surface, mathematically speaking, the paper origami does not consist of a single curved line.

Folding action of the paraboloid

Z

di

st

an

ce

An interesting observation is that, as the paraboloid is released from its folded form, not only does it expand on the y axis, it also expands on the z axis. Folding of the paraboloid can also be achieved by twisting the side edges in the opposite directions. This shows that the paraboloid has exciting potential as an expandable and retractable material.

Essentially, the release of the paraboloid from its folded form increases the length of l, which reduces the size of the angle θ.

Z

di st an ce

X

Z

di

st

an ce

Z

Y

5

Digital Model

CLOCKWISE FROM LEFT 1. Perspective 2. Elevation (front) 3. Elevation (right) 4. Plan 6

DIGITAL MODELLING WORKFLOW

1. Draw two 20cm long perpendicular lines on the front viewport that correspond to the paper edge length on the front viewport.

2. Move one of the lines to the distance of the width of the paraboloid.

3. Apply a normal loft between the two lines.

4. Use a point grid surface domain number command to create a panel of 16x16 based on the lofted surface.

5. Select points alternately based on the edges.

6. Transform the selected points into the measured position.

7. Create a mesh with the points using the Panel 2Dâ&#x20AC;&#x2122;s dense configuration. *Screen captures arranged from left to right, top to bottom

7

Sketch Model

THE SMOOTH ORIGAMI PARABOLOID Based on the findings in the analysis section, I have endeavored to reconfigure the origami paraboloid by exploiting the property of paper becoming three-dimensional when angles in a particular point exceed 360o.

This time, I mimic the property of the zigzags in the origami paraboloid by joining four obtuse angles on four triangles together to form a 3D surface.

FROM TOP: 1. I first plotted a pattern of three adjacent triangles with sides of 5cm and 110o center angle.

2. The pattern was replicated adjacently for 5 times on an A4 paper to generate a continuous surface with the minimum amount of surfaces that are taped together to increase structural integrity.

3. The pattern was cut from the paper, together with five identically-sized triangles from another paper.

The edges were joined together using tape and the surface emerged from 2D to 3D.

8

The finished object represents 5 adjacently connected hyperbolic paraboloids. Ideally, the shape should present itself as a union of completely smooth planes, but due to the differences of the strength of the joined and taped edges, the taped edges are not as smooth as desired. Nevertheless, this prototype shows the potential of utilizing the flexible nature of paper and connecting angles of more or less than 360o to create complex geometry. The angles of the obtuse/ acute angles can be manipulated to influence the fold on the paper.

9

Sketch Design #1 ORIGAMI SLEEPING BAG What is your idea? [Maximum 5 key words]

LOWER EDGE

UPPER EDGE TORSO

PLAN

How does this respond to your personal space?

PERSPECTIVE VIEW

10

The sleeping bag creates personal space for the wearer by covering the top torso by shielding view of the top torso. The size of the bag responds to the increasing size of personal space needed by the wearer as it moves higher in the torso.

Sketch Design #2 PORCUPINE HYPERBOLOID COVER What is your idea? [Maximum 5 key words]

ELEVATION (FRONT)

ELEVATION (LEFT) How does this respond to your personal space?

MATERIAL SYSTEMS

The neck is a highly private region area for humans, yet they move a lot. To ensure comfort and protection, the skin of the porcupine is replicated using paper hyperboloids. The sharp edges deters physical contact, while paper provides flexibility for neck movement.

11

Sketch Design #3 HYPERBOLIC PARABOLOID TENT BED What is your idea? [Maximum 5 key words]

PERSPECTIVE VIEW How does this respond to your personal space?

PLAN

12

ELEVATION

The hyperbolic paraboloid can be stretched in 4 points with an elastic material to generate a hyperbolic paraboloid shape. Here, the upper corners of the paraboloid shields sight of the sleeperâ&#x20AC;&#x2122;s body from the side, hence creating personal space for the sleeper.

13

Appendix

14

15