Applied Mathematics Euclidean Geometry

APPLIED MATHEMATICS

Computational Geometry & FabricationEuclidean Geometry

Written by Yuliya Zavolunova with Anne Leonhardt

The City University of New YorkNew York City College Of TechnologyArchitectural Technology Department

This material is based upon work supported by the National Science Foundation under Grant Numbers 1104254 and 1141234.

Any options, findings, and conclusions or recom-mendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Schience Foundation.

Department ofArchitectural Technology

History

Geometry: “earth measurements”

Throughout history, both the Egyptians (2000-500 B.C.) and Babylonians (2000-500B.C.) demonstrated an early knowledge of Geometry. Greek philosophers/ math-ematicians such as: Thales, Pythagoras, Plato and Euclid studied a combination of Egyptian and Babylonian geometry, creating the first formal mathematics of any kind by organizing geometry with rules of logic.

Euclid“Father of Geometry”

mid 4th century BC-mid 3rd century BC

Euclid’s “the Elements” •Wrote a mathematical and geometric trea-tise consisting of 13 books.•The 13 books cover Euclidean geometry and the ancient Greek version of elementa-ry number theory. •Also includes an algebraic system that has become known as geometric algebra.• Euclid did not discover most theorems, he is generally credited with arranging them in a logical manner

History

GEOMETRY “The power of mathematics is often to change one thing into an-other, to change geometry into language” -Marcus du Sautoy

“There is no royal road to geometry” -Euclid

Geometry by Definition

Geometrical Shapes

Geometry: A branch of mathematics concerned with questions of shapes and their properties (size and position). It is the study of fig-ures in a space of a given number of dimensions and of a given type.

Geometry by Definition

Explicit: Discrete/Directly DefinedEx: Meshes (explained later on) and polylines are explicit because they are defined planes and vertices that are based on their relation-ship.

Implicit: Implied by a Function or other geometry. A function defined by an equation of the form f(x,y)=0Ex: NURBS curves (explained later on) are implicit because they fol-low a specific funtion (determined by domain and range).

*funtion: a rule that certain numbers as inputs assigns to each a definite output. Functions can be represented by ta-bles, graphs, formulasd and descriptions in words.

Geometry by Type

•Points: are an exact position or location on a plane surface. •Curves: a line or outline that gradually deviates from being straight for some or all of its length.•Surfaces: a continuous set of points that have length and width but no thickness.•Meshes: represent (a geometric object) as a set of finite el-ements for computational analysis or modeling.

Grasshopper

Applying These Concepts:

A rule based (parametric/ associciative geometry) visual pro-graming language such as Grasshopper 3D utilizes many geo-metric concepts, which are important to best utilize and learn the power of the programing language.

Grasshopper 3D is built off of Rhino3D, a nurbs-based mod-eling software and allows the user to model geometry by con-structing a series of input rules that work with rhino com-mands. These series of imputs employ parameters, which the user can easily adjust. Grasshopper is optimal for creating mul-tiple iterations (repetitions) and analysis of a form.

The following pages of this primer will use some examples in Grasshopper 3D that give examples of the uses of geometric concepts.

POINTSPoints are represented by an ordered set of numbers called coor-dinates, most likely Cartesian in nature.

Points

•Points are one of the most basic geometrical elements.•A point represents position only. Contains no length, width or depth.•Points typically serve as the underlay for generating more complex geometric types.•Points are also easily generated from, thus dependent upon, other more complex geometric types.

Points in Grasshopper

By using the point component in grasshopper, you can choose any location on the x,y coordinate system. The point component is then used as the basis to create a circle with a specified radius.

COORDINATE SYSTEMSIn geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the posi-tion of a point or other geometric elements in space

2D Cartesian Coordinate System: (x,y)

(0,0)

3D Cartesian Coordinate System: (x,y,z)

-x x

-y

y

-z

z

Unit Circle

•A Unit Circle is defined as having a radius of 1

Coordinate points on the axis of the unit circle are: (1,0), (0,1), (-1,0), (0,-1)

Degrees °

•Degrees are used to express directionality and angle size.

Radians

•Degrees can be represented as Radians (pure measure based on the radius)•Radian is the standard unit of angular measure

Convertions:

•Degrees to RadiansDegree= (Radians) x (1800/π)

•Radians to Degree:Radian= (Degrees) x (π/1800)

r=√x2+y2

y

-y

yr

θ

y=r sinθ

•Polar coordinates function in Radians •Point (p) is defined by its radius (r): length away from origin and θ the rotation from 0 in the positive x direction completing a full circle (360 degrees). •Polar to Cartesian conversion using Pythagorean theorem:

x=r cosθ

θ=tan-1(y/x)

x-x x

origin(0,0)

positive x rotation

P (r, θ)

2D Polar Coordinate System

2D Polar Coordinate System

The purpose of the polar coordinates is to represent curves that have symmetry or spiral about a point.

r=θ, 0 ≤ θ ≥ 4πr=sin(4θ), 0 ≤ θ ≥ 2πr=sin(3θ), 0 ≤ θ ≥ πArchimedean SpiralRose Curve Rose Curve

θ

z

•Cylindrical coordinates are obtained by replacing the x and y coordi-nates with the polar coordinates r and θ and leaving the z coordinate unchanged. •The relationship between Cartesian and Cylindrical coordinates:

Cylindrical Coordinate System

-x

x-y

y

-z

z

r

y=r sinθFrom Cylindrical to Cartesian:

From Cartesian to Cylindrical:

x=r cosθz=z

z=ztanθ=(y/x)r=√x2+y2

CURVESA curve is a topological space*, a set of points which, near each of its points, looks like a line, up to a deformation.

*topological space: a set of points, along with a set of neighborhoods for each point, that satisfy a set of axioms relating points and neighborhoods

Curves

•Curves are defined generally as a set of connected points.•Curve properties include: length, direction, degree, tangen-cy, curvature, points, etc.•Straight lines are considered as Curves in Rhino.•Curves (excluding interpolating curves) can be generated from and controlled by a Control Polygon made of Control Points.•Control Polygons controls the shape of the curve•Control Points determine the shape of the curve

Control Points

control point

• Control points are coefficients of NURBS basis func-tions. Sometimes also called control vertex or node.•Control points are used as “grips” on objects such as curves, surfaces, lights, hatch boundaries, and dimensions and cannot be separated from their objects

Bézier Curve & B-Spline

•Bézier Curves are parametric curves are used in mod-eling smooth surfaces mainly in computer graphics. Used in Programs such as: Adobe Photoshop and Illus-trator. •They are represented through equations which express the coordinates of the points of the curve as a parame-ter (function of variable).•Bézier Curves have interpolates endpoints and do not interpolate middle control points.

•B-Spline is defined as the number of control points and the degree of the curve. •B-Spline does not interpolate ANY control points.•B-Spline curves require more information: degree of curve and a knot vector.•Offers more control and flexibility than Bézier Curves.•Used in Programs such as Rhinoceros 3D.

•Non-Uniform Rational Basis Spline is a mathematical representations of 3D geometry that can accurately describe any shape from a simple 2D line, circle, arc, or curve to the most complex 3D form surface or solid. It offers great flexibility and precision for handling both analytic (shapes defined by common mathematical formulas) and free-form shapes.•NURBS based programs create highly smooth surfaces as opposed to other software where smoothness is limited.

NURBS Curve

Interpolating Curve

Interpolating Curves: Parametric curves that are constrained to pass through the control points that specify them.

NURBS Curve: Degree

Kink: a point where a curve dramatically changes direction.-can also happen at a point where a curve dramatically changes the amount it curves.

Lines and polylines are degree 1 NURBS curves

Circles and ellipses are ex-amples of degree 2 NURBS curves. They are also rational or non-uniform curves

Free- form curves are usual-ly represented as degree 3 NURBS curves

Degree of curves come from powers of polyno-mialspolynomial examples: Linear: x+2quadratic: x2-3x-6cubic: x3+6x2-4

First Degree Curve:

Third Degree Curve:

NURBS Curve: Degree

Control points have an associated number called a weight. With a few exceptions, weights are positive numbers. When a curve’s control points all have the same weight (usually 1), the curve is called non-rational, otherwise the curve is called rational.

Weight of a Curve

Tangent Line to a Plane Curve at a Given Point (p)

p

Normal Direction of Curves

•Direction for curves are determined originally by the start and end points specified when they are drawn.

Curve 1 Curve 2

Curve Direction- Grasshopper

By isolating points on a curve and us-ing them as the anchor points in a Vec-tor display component, you can see the curve direction

Geometrical Continuity of a Curve

•A curve or surface can be described as having an infinite measure of smoothness•Order of continuity: - Position (G0) - Tangency (G1) - Curvature (G2) - G3 - G4

Geometrical Continuity of a Curve (G0)

•Positional Continuity (G0): measures location only. •Established when endpoints of two curves or edges of two surfaces share a position in space•Kink between two shapes can be seen.

curve 1 curve 2

-shared position (endpoints)-kink

Geometrical Continuity of a Curve: Position (G0)

Here two curves are bridged, using a "0" in the continuity input. This cre-ates a position blend which simply draws a straight line from the end of each point.

kink

kink

Geometrical Continuity of a Curve (G1)

•Tangent Continuity (G1): measures position and curve direction at the ends. Two curves not only touch, but they go the same direction at the point where they touch. The direction is determined by the first and second point on each curve. If these two points fall on a line, the two curves are tangent at the ends.•Tangent continuity can be derived through the fillet command.•Contains slightly visible kink.

curve 1

curve 2

-tangent continuity between two curves (fillet command)

Geometrical Continuity of a Curve: Tangency (G1)

Here two curves are bridged, using a "2" in the continuity input. This creates a tangen-cy blend which blends the two curves according to their tangents.

Geometrical Continuity of a Curve (G2)

•Curvature Continuity (G2): between two curves measures position, di-rection, and radius of curvature at the ends. The curves not only go the same direction when they meet, but also have the same radius at that point.•Transition of curvature continuity can be seen by using the blend command in rhino.•Contains no visible kink.

curve 1

curve 2

-curve continuity between two curves (fillet command)

Geometrical Continuity of a Curve: Curvature (G2)

Here two curves are blend-ed, using a "1" in the con-tinuity input. This creates a curvature blend which draws a curve based on the curvature of the curves end points. This provides the smoothest continuity.

Geometrical Continuity of a Curve (G3 & G4)

•G3 continuity adds a third requirement: planar acceler-ation. Curves that are G3 continuous touch, go the same direction, have the same radius, and that radius is accel-erating at the same rate at a certain point. G3 continuous curves have equal third derivatives.

•G4 continuous curves have all the same requirements as G3 curves, but their curvature acceleration is equal in three dimensions.

Geometric TransformationA geometric transformation is a function whose domain and range are sets of points. Geometric transformations are often 1 to 1 functions (one imput is paired with 1 output), so they have inverses. We can manipulate 2D and 3D geometries through var-ious transformations such as: displacement, affine, euclidean, similarities, etc...

Detailed description will be provided on affine and euclidean transformations on the following pages.---???

Affine Transformsation: transformation that preserves points, lines and planes, as well as parallel lines (does not preserve distance be-tween points and angles). -Scaling (uniform): enlarges or shrinks objects by a scale factor that is the same in all directions.-Scaling (non-uniform): obtained when at least one of the scailing factors is different from the others.

Grasshopper 3D provides a variety of affine transformation.....

Affine Transformation

Euclidean Transformsation: a rigid transformation (isometric) that pre-serves both the distances between every pair of points (length) and their angles. -Rotating: circular movement of an object arounds a fixed center (or point) of rotation (rotational axis). •Rotating 3D •Rotating about an Axis-Move: translating an object along vector(s) x, y or z (direction)

Euclidean Transformation

Creating the floor profile requires the Move component which trans-lates the original curve in a specific direction. Through the Vector XYZ component we translate the curve in the z-axis. Parameters for the z-axis are set by the Range component which allows us to move the original curve multiple times to create desired amount of floor levels. Grasshopper works of the units of Rhino so the Multiplication component is used to convert inches to feet. This allows us to create the maximum height of the building elevation.

The Scale NU (non-uniform) component, in this case, is used to independently scale the floor profile curves in the x and y axis. The Area component denotes the centroid (cen-ter point for each geometry) to be transformed. Because we use a range with the same number of steps as the move we create equal length lists and can transform each curve one time.

Rotating the floor profile can be achieved through the Rotate component. The geomoetry are the previously scaled curves. The angle of rotation is determined by the Series component creating a list of values with a specific starting point, a step number (val-ues between outputs) and the count number (number of output values). The degree of rotation (Rotate component) is determined by the Series' output. Each step is 8.7 degrees with a count of 42, one per floor profile.

CONICCurve generated by the inter-sections of a plane with one or two nappes of a cone. The point where the two tips meet is called the vertex. May be defined as a plane algebraic curve of degree 2.

*RHINO 5: Conic Curve Command

vertex

Degenerate Conic

Point Line Two Intersecting Lines

A degenerate conic is a conic (a second-degree plane curve, the points of which satisfy an equation that is quadratic* in one or the other or both variables) that fails to be an irre-ducible curve.*quadratic: a polynomial function of degree 2

Conic Section by Type

Parabola Ellipse and Circle Hyperbola

Parabola

Parabola is formed by the intersec-tion of a plane and a single cone

Ellipse and Circle

Ellipse is formed by the inter-section of a plane traveling at an angle to the base of the cone.

Circle is formed by the intersec-tion of a plane traveling parallel to the base of the cone.

Hyperbola

Hyperbola is formed by the intersection of a plane and a double cone

DOMAINThe domain of a function is the set of all possible inputs values, which produce a valid output from a particular function.-Domain of a curve is a value between 0 & 1, where 0 is the start of the curve, and 1 is the end. The domain is influenced by the curvature of the curve.

NURBS Surface Domain

SURFACESSurfaces define a two dimensional Parameter Space that we can move along (u,v) to query properties at that location. It is sepa-rate from the world axis system (x,y).

•Surfaces are defined as a set of connected points, related in two dimensions, typically done through curves.•Surfaces inherit all properties from the Curves that create them and have additional properties: area, surface curva-ture, etc.•NURBS Surfaces can be generated from and controlled by a Control Cage/Mesh made of Control Points.•Using NURBS Surfaces reduces the memory consump-tion when storing geometry•Can be described implicitly* or parametrically*

*implicit: implied by a function or other geometry.*parametric: one of a set of independent variables that express the coordinates of a point.

Surface

Normal Direction of a Surface

•For surfaces the normal is a direction that points toward what you can think of as “outside” or “up.” For closed polysurface (cone, cylinder, box, etc.) or single-surface solids (sphere, torus), the normal always points “out.” Howev-er, on an open surface or polysurface, the direction of the normal depends on how it was created and can seem arbitrary.

Solid 1

normal surface flipped surface

Surface 2

•If the surface has been flipped, a small blue line indicates the “natural” normal.

Extrusion Surface

Curve

Extruded surface

from a curve

Tangent Plane to a Surface at a Given Point (p)

Rotational Surface: Rotated about a defining axis.

Ruled Surface: A ruled surface can always be described (at least locally) as the set of points swept by a moving straight line. For example, a cone is formed by keeping one point of a line fixed while moving another point along a circle.

Rotational & Ruled Surface

The Extrude component extrudes all the curves in the z-axis through the Unit Z component (vector). The mag-nitude of the vector of the extrution for the floor slab (thickness) is determined by a specific value which is obtained from the Number Slider component. These extruded surfaces follow a single direction of curvature and are generated as ruled surfaces. Creating a planar surface in openings can be achieved through the Cap Holes component.

Extrude Cap

Lofted and Swept Surfaces

•A surface that can be generated from two curves by translat-ing either one of them parallel to itself in such a way that each of its points describes a curve that is a translation of the other curve.•Surface can be generated without rotating or flipping their orig-inal components. •Representative of lofted and swept surfaces created from curves.

Lofted Surfaces in Grasshopper

The Loft component creates a surface by us-ing the rotating curves. A lofted surface has more degrees of freedom then a simple extru-sion.

FULL GRASSHOPPER DEFINITION GOES HERE

Network Surface

Network Surface: Creating a surface from a network of crossing curves. This process offers more control than other Surface commands.

Create curves which will be used to form the surface (u,v values).

Select curves based on desired u,v direction of the surface.

Final Surface

Weight of a Surface

MESHESMeshes have a different set of properties than NURBS curves and surfaces: face description, edge description, vertex descrip-tion, topology, etc.

Meshes

•Meshes are defined as a set of points (vertices) arranged into basic elements (faces).•Meshes can be used to generate faceted geometries to define built elements or approximate curvature.•Planar facets of a mesh can only approximate a curved shape.•Meshes in rhino are used in Rhinoceros 3D to texturemap, render and 3D print

Meshes

Meshes

•Polygon meshes can be created from a NURBS surface or polysur-face. The more polygons a mesh contains, the smoother it is. •Meshes in Rhino consist of triangles and quadrilaterals•Rhino creates triangular and quadrilateral meshes for various file format exports•Rhino is not optimized for mesh modeling: plug-ins such as Weav-erBird and T-Splines are used to provide flexibility within Rhino for mesh modeling.

mesh refinement

Quadrilateral Triangle

Mesh Faces