An algorithm that translates intrinsic equations of curves into intrinsic procedures of these curves

Uzi Armon
Bar-Ilan University - The Jorden-valley college
Department of Math and Computer Sciences
Jorden-valley, 15132, Israel
Tel.: 972-6-6989240


The intrinsic algorithm, demonstrated here, links classical mathematical topics with turtle geometry. It uses the turtle's fundamental property that it always looks in the direction of the tangent to the curve. Thus, it "directs" the turtle to proceed in a "natural way" along the curve and at the same time drawing it. That algorithm link turtle mathematics with some more standard mathematical fields. This link is formed by means of specific translations between the mathematical formalism and the Logo formalism. The main aspect of this translation is that it is from a declarative and algebraic formalism (curve equations) to a procedural and computational one (curve procedures).

This algorithm, translates each intrinsic equation of a curve to intrinsic turtle procedure to draw the same curve. This algorithm inputs an intrinsic equation (or function) of a curve of the type r = r (f ) (where r is the radius of curvature and f is the inclination angle), and "outputs" a new procedure whose "action" part is a simple step of the form RIGHT 1 FORWARD <r >. This algorithm can be represented by another general procedure INTR_GRAPH :FUN :FIRST.PHI :LAST.PHI , which produces a graphical representation of the same curve. Thus we can use the intrinsic equation of a cycloid r = sin(f ) to draw it. In the same way we can use the intrinsic equation of an Astroid (an example of the Hypo-Cycloid curves) r = sin(2f ), or that of a Cardioid (an example of the Epi-Cycloid curves) r = sin(f /3) to get their graphical representations.


turtle geometry, intrinsic equation, intrinsic procedure, cycloidal curve.


1 Demonstration of the algorithm

This section presents the algorithm that translates an intrinsic equation of a plane curve into an intrinsic procedure that will draw the same curve. The algorithm is exemplified here by the Cycloid (Fig.1), because its intrinsic equation is simple.

The usual mathematical definition of Cycloids uses parametric equations (such as (12) or (17) below). Its physical definition says that it is the path created by a point of a circle that rolls clockwise (without slipping) on a straight line. Here, we prefer the definition by its intrinsic equation. This equation has two intrinsic variables: the tangent-angle f , and the curvature-radius r :
(1) r (f ) = a × sin f
(the parameter "a" determines the size of the Cycloid).

The equivalent of any such equation, as a mathematical representation of a curve, is a computerized procedure that contains, in our case, a simple-action-step. A simple-action-step (emphasized below in the CYCLOID procedure) is built from two commands in turtle graphics: turning right (or left) by: right <angle> and advancing by: forward <step>. The equivalent procedure is:

if (:TOTAL > 180) [stop]
right 1 forward sin :TOTAL
Fig.1: Cycloid by: CYCLOID 1

For example, if we were replacing in the above procedure the expression sin :TOTAL (where :TOTAL is the value of the variable TOTAL) by any constant number C (like 1), then that procedure, with the new simple-action-step, will draw a circle with a radius which equals to (180/p )× C.

The CYCLOID procedure is an approximate intrinsic representation of the Cycloid [8]. The length of its step: forward sin :TOTAL is similar to the intrinsic equation (1) of the Cycloid for a=1. This similarity can be generalized, in such way that for any intrinsic curve equation r = r (f ) there is a suitable procedure in turtle graphics which causes the turtle to draw approximately the same curve. This procedure contains a repetition of a simple-action-step, in which the turn is by 1° and the stepping forward is r (f ). Now, we will discuss the two main basic concepts here - intrinsic equations (in math) and intrinsic procedures (in Logo programming).

2 Intrinsic equations

2.1 Intrinsic variables of curve

This section presents the mathematical background for the article and it explains the concept "intrinsic equation" of a curve [7,p.5]. Such an equation is written by intrinsic variables [6,p.24], like: arc-length s, tangent-angle f , or curvature-radius r , instead of the usual extrinsic variables x and y. These variables do not depend upon any external reference system but rather upon an internal reference point of the curve under discussion. That point is the origin, and the tangent of the curve at the origin is the reference axis of the intrinsic system (of the curve). And then, for each point on the curve, there are the following respective values of the intrinsic variables:

(2) r = ds/df { = (ds/dt) / (df /dt) }

Although, the value of each intrinsic variable depends upon the origin, the increment between any two points on the curve is constant, and therefore the shape of the curve doesn't depend upon the choice of the origin. One connection, between the intrinsic variables and the usual extrinsic Cartesian variables, will be exemplified below. Later on, various types of intrinsic equations, will be discussed.

In case of a curve with explicit Cartesian representation y=f(x), where the reference axis is parallel to the x-axis, the tangent-angle has the standard form:

(3) f = arctan f’(x)

Expression (3) is applied by another algorithm in turtle graphics, called "Graphical integration" [3]. That algorithm supplies each explicit elementary function y=f(x), with a graphical representation of its indefinite integral F(x) = ň f(x)dx. Thus, it is similar to the formal integration, which might supply an explicit elementary function with a formal description of its indefinite integral. The algorithm is based upon the fact that the derivative f'(x) at each point x of the function y=f(x) is its slope there, and that the arctan function provides the inclination angle for every slope. This is also the same angle to which the turtle's head must be set so as to draw the graph of the function y=f(x). Now, by composing arctan with the function f itself, rather than with its derivative f', one obtains the graph of its primitive function, that means the graph of its indefinite integral ň f. The main advantage of the graphical integration, is that it works also on functions, where their integral has no explicit elementary description, like the function: y = for any constant a.

2.2 Intrinsic equations

An intrinsic equation (called also a natural equation) of a curve, connects 2 of the 3 intrinsic variables: s, f and r . From extrinsic point of view it is a differential equation with a parameter t or with the variables x and y. In the mathematical literature there are names for 2 of the 3 possible types of these equations [13,p.126].

A Whewell equation is an intrinsic equation with the variables s and f . It is called after W. Whewell that first studied this topic in 1849 [11,12], and also invented the name "intrinsic equation". For example, equation (4) is the Whewell equation of the Cycloid:

(4) s = s(f ) = 4r× sin f

A Cesáro equation is an intrinsic equation with the variables s and r . It is called after the name of E. Cesáro, who studied this topic in the year 1900 [5], about 50 years after Whewell, probably without knowing of his predecessor, and called it "natural equation". For example, equation (5) is the Cesáro equation of the Cycloid:

(5) s2ý + r 2 = 16r2ý

An intrinsic equation of the "third type" is an equation with the variables f and r , like (6) for the Cycloid:

(6) r = r (f ) = 4r× cos f

Any equation of that type can be translated by the algorithm presented here into intrinsic procedures in turtle graphics. That "Turtle equation" had no name, and also was not interesting, except for one case, according to Whewell, in which Euler, in an article in 1736, used it in order to prove a theorem of Bernoulli [12].

The transitions among the three types of intrinsic equations are generally done by means of derivation or by integration, like the transitions between Whewell equations and Turtle equations.

3 Intrinsic procedures

This section presents the concepts associated with the term "intrinsic" such as intrinsic axis system and intrinsic procedure.

It is possible to draw a graph of any function in two essentially different ways: In intrinsic (and dynamic) axis system and in an extrinsic (and static) axis system. Drawing a graph by means of commands, like forward and right causes a "natural" movement of the turtle on the screen by means of stepping it forward (or backward) and turning it right (or left). In this case the reference system is attached to the turtle and moves with it everywhere. Thus, a procedure that includes only commands "to steer the turtle", is called an intrinsic procedure.

In addition, there are extrinsic commands such as setpos or setheading, that can bring the turtle to any point (x,y) on the screen or rotate its head to any direction h in relation to the North axis. The function heading returns that direction. The CYCLOID, described in the previous section, has another representation that combines the two ways: repeat 180 [right 1 forward sin heading].

Turtle graphics is a reference system that is based upon two variables: length of the step (for forward) and angle of the turn (for right). Length of the step is approximately the difference of the arc-length, and the angle of the turn is approximately the difference of the tangent-angle. The turtle reference system, for these variables, is determined by the original state (x, y, h) of the turtle, where the origin is determined by the location (x,y) of the turtle, and the reference axis of the reference system - by the direction h of its heading. For any curve in turtle graphics, only its position on the screen is determined by the original state of the turtle, while its shape and its other (intrinsic) properties are conserved under the rotation and the translation operations. Therefore, turtle graphics is an intrinsic reference axis system.

A formal description of functions commonly uses equations, such as Cartesian, polar, or parametric. The computer adds a new viewpoint to the concepts of functions. For example, it enables describing them by means of procedures instead of by equations.

In addition, turtle graphics demonstrates the intrinsic and dynamic representations that might arise mathematical, computerized and also educational interest. Following, we examine the "existence question": Which plane-curves have intrinsic representation in turtle graphics? Here we give a partial answer - any curve that has an intrinsic equation. The following algorithm provides every such curve with a turtle representation by translating its intrinsic equation to an intrinsic procedure, to draw the same curve.

4 Description of the algorithm

This section explains the mathematical connection between turtle graphics and an intrinsic system. This connection has to overcome some natural difficulties that arise when one adapts a continuous mathematical system to a discrete computerized system. It will be done by an intrinsic procedure with an input of which is an intrinsic equation of a curve, and then, it will draw the same curve.

4.1 Explanation of the algorithm

The step length of: forward sin :TOTAL in the intrinsic procedure CYCLOID from section 1 reminds us of the intrinsic Whewell equation: s = sin f of the Cycloid equation (4) for 4r=1. It is easy to see that, in addition to the "intrinsic" feature which is mutual to them, the sinus function plays important and similar role in the two representations. The sinus affects the arc-length s in the Whewell equation, and also the length of the step of forward that represents a segment D s of the curve. In addition, there is also similarity in the angular input of the sinus. The variable f represents the tangent-angle of the curve, and the variable TOTAL represents the angle of the total turn, which is the direction of the turtle head in relation to its original direction. When the turtle draws any "continuous" curve it progresses according to the tangent of this curve. Therefore, the angle of the total turn is exactly the tangent-angle in relation to the original direction of the turtle.

As a matter of fact, the "nice" similarities are somewhat misleading. The command right represents the difference D f of the tangent angle and not the angle f itself. The same with the command forward that represents, in a similar way, the difference D s of the arc-length and not the arc-length s itself. Thus, the turtle commands represent the differences or the increments of the intrinsic variables. It is possible to represent these increments by difference equations. Equation (2) has the differential representation: ds = r (f ) df , and the respective difference equation is:

(7) D s ş s(f +D f ) - s(f ) = r (f ) × D f

In turtle graphics the step length suits the D s, and the angle of the turn suits the D f . Therefore, the scheme of the simple-action-step, which suits (7), is:

right <D f > forward <D s>. When the angle of the turn of the turtle is one degree (D f = 1° ), then the length of the step is:

(8) D s = r (f )

and the scheme of the simple-action-step will be: right 1 forward <r (f )>.

Thus, it is possible to translate each Turtle curve equation: r = r (f ) to a simple-action-step in turtle graphics, such that repetitious execution of it will direct the turtle to draw the same curve. By that, the main theorem, that connects between intrinsic equations and intrinsic procedures, was proved:

Theorem: For each intrinsic Turtle equation r =r (f ) of a curve, there is a respective procedure in turtle graphics which includes a repetition of the simple-action-step right 1 forward <r > that directs the turtle to draw approximately that curve.

Somewhat similar formulation of this theorem appears in Whewell's article [10,p.659] about 130 years before Papert [10] created turtle graphics in Logo:

Using the common notation we have ds = f'(f ) df . The curve may be constructed approximately by taking small finite differences of f , and determining the corresponding rectilinear elements of the polygon or approximate curve, by the equation D s = f'(f ) D f .

4.2 Accommodation of the algorithm to mathematics

By deriving the Whewell equation of the Cycloid according to f , one gets r as a function of f , for 4r=1:

(9) r = ds/df = cos f

This expression, that is not identical to that which appeared in the CYCLOID procedure, exemplifies two technical difficulties of translating a mathematical language into a computer language.

One difficulty here emerges because the angles in turtle geometry are measured by degrees and not by radians, and thus the derivative of sin function is: (p /180)× cos. Also, the angles are measured there relative to the North axis and clockwise, that means, in the negative mathematical direction. The first difficulty affects only the intrinsic equations with trigonometric functions, and only up to the constant (p /180).

The second difficulty is solved by the "accommodation operator": a --> 90-a . That operator converts the function cos f to sin f , and therefore, in the CYCLOID procedure, exemplified at the beginning of the article, the following simple-action-step: right 1 forward sin :TOTAL represents the equation:

(10) r = cos (total)

And the Cycloid equation:

(11) r = a cos f

represents the simple-action-step: right 1 forward (:a * sin :f ).

It stems from the main previous theorem, that "the accommodation operator" enables constructing a procedure, that represents the algorithm including in that theorem:

if (:PHI > :LAST_PHI) [stop]
right 1 forward (run :FUN)

The parameter PHI is the counter of the loop, that runs from 1 up to the last value - LAST_PHI, which depends upon the curve (or upon its intrinsic equation). For example, to get one branch of the Cycloid, the value of LAST_PHI is 180, and for the Cardioid, that will be demonstrated in the continuation, LAST_PHI is 540. The first parameter FUN, contains the intrinsic equation of type: r = r (f ). Then, for example, the instruction: INTR_GRAPH [sin :PHI] 1 180 will draw a Cycloid (Fig.1), like the output of the procedure CYCLOID at the beginning of the article. In the same way, the next instruction: INTR_GRAPH [0.001 * :PHI] 1 1000 will draw a Spiral, named Involute of circle (Fig.4.1), which is described as the third example in the next section.

5 Demonstration of the translating algorithm

This section presents several examples of curves that have a representation by means of intrinsic equations. The main purpose is to exemplify the general idea, of translating intrinsic equation of a curve into an intrinsic procedure. Additional goal is to demonstrate the properties of intrinsic representations, and mainly their simplicity and their power.

5.1 Cardioid

The Cardioid is a curve with a shape of a heart (Fig.2.1), that belongs to the family of the Epi-Cycloids. A curve of that family describes the path of a point of a circle that rolls without slipping upon a constant other circle [9]. A Cardioid is created when the radii of the two circles are equals. If the ratio between the radius of the constant circle and the rolling circle is a natural number m, then this is also the number of the cusps of the curve (in case of the Cardioid: m=1). The Epi-Cycloid with m=2 is called Nephroid and indeed it has two cusps (Fig.2.2).

The extrinsic representation of a Cardioid is derived from the general parametric representation of Epi-Cycloids, which is:

(12) x = (k+r) × cos(t) - r × cos[t× (k+r) / r]
y = (k+r)
× sin(t) - r × sin[t× (k+r) / r]

where the parameter t describes the "rotation angle" of the rolling circle, and the variables k and r describe the radii of the constant circle and the rolling one, respectively. Those equations can be simplified for the Cardioid, in which r=k, up to translation by k in the x-axis:

(13) x = 2k × (1 - cos t) × cos t
y = 2k
× (1 - cos t) × sin t

All Epi-Cycloids have a general intrinsic representation, which is equivalent to (12), by the Whewell equation:

(14) s = a × sin (f / n)

The parameter n is a natural number that determines the type of the Epi-Cycloid. In case of the Cardioid: n=3, and for the Nephroid: n=2. The parameter a determines the size of the curve and in case of the Cardioid: a=8r(=8k), and in case of the Nephroid: a=6r(=3k). In case of a simple Cycloid, with n=1, there exists: a=4r [7,p.195].

The derivative of Whewell equation of the Cardioid s = a × sin f /3 (f in degrees) is the intrinsic Turtle equation:

(15) r = a × (p /540) × cos (f /3)

And the intrinsic Turtle equation of the Nephroid (with n=2) is:

(16) r = a × (p /360) × cos (f /2)

By substituting it in the procedure INTR_GRAPH, with the help of the "accommodation operator", we'll get the intrinsic procedure CARDIOID (for a=540/p and thus r=67.5/p ), and by executing it, the turtle will draw the Cardioid curve (Fig.2.1):

if (:PHI > 540) [stop]
right 1 forward sin (:PHI / 3)

1. Cardioid by: CARDIOID 1 2. Nephroid

Fig.2: Epi-Cycloids

We can get the same drawing by executing the general procedure:

INTR_GRAPH [sin :PHI / 3] 1 540. And the intrinsic description of the Nephroid (Fig.2.2) will be: INTR_GRAPH [sin :PHI / 2] 1 720.

5.2 Astroid

The Astroid is a curve that is similar to a rhombus, built from arcs instead of line segments (like Fig.3.1). This curve belongs to the Hypo-Cycloids, which are created by a point of rolling circle without slipping inside a constant second circle. The Astroid is produced when the radius of the constant circle is 4 times big as the inside rolling circle. In general, if the ratio between them is a natural number m, then it is also the number of the "cusps" of the curve (in case of the Astroid: m=4). The Hypo-Cycloid with m=3 is called Deltoid, and indeed it has 3 cusps (Fig.3.2).

The general representation of Hypo-Cycloids by means of parametric equations is:

(17) x = (k-r) × cos(t) + r × cos[t× (k-r)/r]
y = (k-r)
× sin(t) - r × sin[t× (k-r)/r]

and it can be simplified for the Astroid by:

(18) x = k × cos3(t)
y = k
× sin3(t)

All Hypo-Cycloids have a general intrinsic representation, like the Epi-Cycloids, by means of a Whewell equation, which is equivalent to (17):

(19) s = a × sin (n × f )

The parameter n is a natural number that determines the type of the Hypo-Cycloid. In the case of the Astroid: n=2, and in the case of the Deltoid: n=3. The parameter a is a constant that determines the size of the curve. In the case of the Astroid: a=r=k/4, where in the case of the Deltoid a=(8/3)r=(2/3)k [7,pp.131-134].

The derivative of Whewell equation of the Astroid: s = a × sin 2f (f in degrees) is the intrinsic Turtle equation:

(20) r = a × (p /90) × cos (2× f )

By substituting it in the procedure scheme INTR_GRAPH, we'll get the intrinsic procedure ASTROID (for a=90/p , and then k=360/p ), and by executing it, it will draw the Astroid (Fig.3.1):

if (:PHI > 360) [stop]
right 1 forward sin (:PHI * 2)

1. Astroid by: ASTROID 1 2. Deltoid

Fig.3: Hypo-Cycloids

A similar drawing will be accepted by executing the general procedure:

INTR_GRAPH [sin :PHI * 2] 1 360. An Intrinsic description of the Deltoid (Fig.3.2) is accepted by: INTR_GRAPH [sin :PHI * 3] 1 180.

5.3 Spiral - Involute of a circle

In turtle geometry it is also possible to get other kinds of curves like the Spirals, and one of them, exemplified below, is the Involute of a circle. This curve has an elegant connection with a familiar procedure from turtle graphics named POLYSPI [1,p.77]. From the viewpoint of the turtle, the drawing of the Spirals is similar to the drawing of circles. Papert [10] elucidates this analogy, by demonstrating the connection between the intuition and the formalism of it:

The spiral gets flatter, "less curvy," as you move out from the middle. ... The curvature of the spiral decreases as it moves outward. To walk in a spiral one could take a step, then turn, take a step, then turn, each time turning a little less (or stepping a little more).

(Papert, 1980, p. 69)

The representation of an Involute of a circle, by means of parametric equations, is:

(21) x = a × (cos t + t × sin t)
y = a
× (sin t - t × cos t)

Its representation, by means of Whewell equation, is:

(22) s = (a/2) × f 2

The transition to an intrinsic Turtle equation, by means of derivation, gives:

(23) r = a × f

The substitution of (23) in the procedure scheme INTR_GRAPH will provide (for a=0.001) an intrinsic representation of this Spiral:

if (:PHI > 1000) [stop]
right 1 forward (0.001 * :PHI)

1. Involute by: INVOLUTE 1 2. Polygon spiral by: POLYSPI 60 1

Fig.4: Involute of circle


The procedure Involute has the same structure as the classic procedure [1] in turtle graphics - POLYSPI (for constant value, like 1, of the parameter ANGLE):

if (:SIDE > 70) [stop]
right :ANGLE forward :SIDE

For large angles, like ANGLE = 60, this procedure draws polygon Spirals (Fig.4.2). For small turn angles, like ANGLE = 1, and progression in an infinitesimal steps, like :SIDE / 1000, POLYSPI represents a mathematical curve - the Involute of a circle.

6 Advantages of the algorithm

The discussion about the advantages of the algorithm, that translates intrinsic equations to intrinsic procedures, will relate also to the advantages of intrinsic representation at all. In the continuation we will present the properties of that algorithm, such as the power of its simplicity; its generalization; illuminating new aspects of turtle geometry; and finally, the applications to dynamic mathematics, which illuminate the mysterious that still exists in this interdisciplinary area.

6.1 Simplicity

The algorithm presented here has a special and powerful simplicity. It is simple to apply just by writing the intrinsic equation of a curve as an input to the procedure INTR_GRAPH, and then, by executing it to get that curve immediately. Also the algorithm has a simple content - a repetition of a simple-action-step, where the right turn is 1° and the length of the step is according to the intrinsic equation under discussion.

Additional aspect is about the simplicity of the intrinsic representation. There are a lot of curves, like the Involute of circle (21), the parametric representation of which are so complex, that they are not studied even in the academic level, while their intrinsic representations are very simple, like for example, the linear equation (23): r =af .

6.2 Generalization

One of the advantages of the intrinsic representation is its ability of generalization. There are families of curves like the Cycloidal curves [4], where any sub-family has its own parametric representation, while the intrinsic representation is common to all (see also [2]). The first two examples in section 5, the Cardioid (13) and the Astroid (18), are specific cases of the generalization of all Cycloidal curves - Epi-Cycloids (12), Hypo-Cycloids (17), and simple Cycloids:

(24) s = a × sin (n × f )

This Whewell equation, similar to (19), represents:

  1. Simple Cycloids for n = 1;
  2. Hypo-Cycloids for 2 Ł n natural, like the Astroid for n = 2, and the Deltoid for n = 3;
  3. Epi-Cycloids for n = 1/m, where m is a natural number, like the Cardioid for n = 1/3, and the Nephroid for n = 1/2.

6.3 Enhancement of the comprehension of turtle geometry

The possibility of drawing such curves in turtle graphics, that have intrinsic equations, enables to inquire their mathematical properties through programming. This is "2-way street", since that connection enables us also to use known mathematical results about the intrinsic equations, in order to improve the comprehension of turtle geometry. For example, the discovery in example (5.3), that the classical procedure POLYSPI to draw Spirals, represents actually the Involute of a circle.

There are several additional problems connected to the mathematical aspects of intrinsic representations, that meantime have no solution. For example, although equation (19): s = a × sin(n× f ) determines the kind of the Hypo-Cycloid, according to the natural value n, yet this connection between this parameter and the number of the cusps of Hypo-Cycloid is still unclear! Another question is: Which mathematical curves are accepted for fractional values of the parameter n? Similar questions exist for the other sub-family - the Epi-Cycloids. Additional question is: Does similar intrinsic representation exist for not simple Cycloids, like the Curtate Cycloid or the Prolate Cycloid?

6.4 Applications to dynamic mathematics

The intrinsic representations have also a physical meaning for the concept of motion, as well as for dynamic drawing of mathematical curves. This may be important to any domain deals with orbital motion like in kinematics (eg. missiles orbits) or in robotics (eg. the orbit of robot arms). Thus, when one plans cycloidal motion, there is no more need to calculate each point of its path by its complex parametric equations, rather, he has to determine the angle of turn and to progress according to the sinus of its total turn angle.

Additional application is to the field of mathematical inquiry of functions. When one wishes to receive a graph of a curve according to its curvature-radius, for example: r = C + sin2f (Fig.5.1), or: r = C / Ö f (Fig.5.2) for any constant C, there is no need to solve such differential equations, or to use numerical analysis methods. Again, as mentioned, after deciding on the desirable approximation (by determining the turn angle of every step) one has to progress according to the value of r in every stage.

1. r = 0.2 + sin2f 2. r = 5 / sqrt (f )

Fig.5: Learning curves by their curvature



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