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Advances in Technology Innovation, vol. 2, no. 1, 2017, pp. 08 - 12 

8 

The Tangent Medial Circles Inside the Region Defined 

by Hermite Curve Tangent to Unit Circle 

Ching-Shoei Chiang 

Computer Science and Information Management, Soochow University, Taipei, Taiwan. 

Received 19 February 2016; received in revised form 08 April 2016; accepted 10 April 2016 
 

Abstract 

The design of curves, surfaces, and solids are 

important in computer aided geometric design 

(CAGD). Images, surround by boundary curves, 

are also investigated by many researchers. One 

way to describe an image is using the medial 

axis transform. Under this consideration, the 

properties of the boundary curves tangent to 

circles become important to design for 2D im-

ages. In this paper, we want to find the medial 

axis transform (MAT) for a special class of 

region, which is bounded by unit circle and the 

curves whose end points is on a the circle, and 

endpoints tangent vectors are parallel to the 

tangent of circle at the end points. During the 

process, we want find the medial circle tangent 

to other medial circle, until we reach the medial 

circle whose center is the center of the osculating 

circle for the point with local maximum curvature. 

There are 4 cases, symmetric/non-symmetric re-

gion with singular point/local maximum curvature 

point, and proposed algorithm for these 4 cases.  

We introduced algorithm for this 4 cases in this 

paper. 

Keywords: medial axis transform, hermite 

curve, computer geometric model-

ing 

1. Introduction 

The 2D image can be stored by many dif-

ferent ways, including its boundary curves, a 

medial axis curves with radius function, union of 

many primitive figures, and so on. There are 

many researchers use different curves to simu-

late different images. For example, Cinque, 

Levialdi and Malizia [1] uses cubic Bezier curve 

to do the shape description. Yang, Lu and Lee[2] 

use Bezier curve to approach the shape descrip-

tion for Chinese calligraphy characters. Chang 

and Yan[3] derived an algorithm to approach the 

hand-drawn image by using cubic Bezier curve. 

Cao and Kot[4] derived an algorithm to do data 

embedding in electronic inks without losing data. 

The boundary curves can be also used to derived 

the offset curves and medial axis of an images[5], 

and also to simulate the nature objects, such as 

flowers[6]. In this paper, we would like to in-

vestigate the cubic Hermite curves, where the 

end points are on a unit circle, and its end points 

tangent line is parallel to the tangent line of 

circles. Using these properties, with more con-

straints on the singular point or maximum cur-

vature at specified parameter, we survey the 

couture of the curves, so that the design of the 

curve may be simplify by giving constraint. 

2. Definition and Theorem 

Let introduce the Hermite curve first. Given 

two points P0 and P1 , with two associated tan-

gent vectors   and V1 , the Hermite curve C(t) 

defined as (See Fig. 1): 

 

 
Fig. 1 Hermite Curve 

To simplify our problem, we assume that the 

end points P0 and P1 are on the unit circle, the 

first point P0 =(-1,0), and the second point P1 

=(-cosθ , sinθ). It’s associated tangent vectors 

parallel to (0,1) and (sinθ , cosθ ). We have the 

following definition for this curve. 

* Corresponding author, Email: chiang@scu.edu.tw 



Advances in Technology Innovation, vol. 2, no. 1, 2017, pp. 08 - 12 

9 Copyright ©  TAETI 

Definition 1: Two points P0 =(-1, 0), P1 

=(-cosθ , sinθ ) on the unit circle with its asso-

ciated tangent vector V0=α0(0,1), V1=α1(sinθ , 

cosθ ) produced a Hermite curve, we call it unit 

circle Hermite curve, denoted H1(t;θ,α0,α1), 

where α0>0, α1>0, 0<t<1, 0<θ<2π. 

Notice that for the general case, we can al-

ways convert the design problem into the design 

problem for H1(t;θ,α0,α1). We can always trans-

late the center of current medial axis circle to the 

origin, scale the radius into one and rotate the 

circle so that the first boundary point is on (-1,0). 

We call this process the standardization of the 

problem. After we solve the problem, produce 

the curve we want, we can always inverse the 

rotate, scale, and translate process, to see the 

designed curves for the original design problem. 

In [7], we have the following two theorems: 

Theorm 1:  Let C(t)=H1(t;θ,α0,α1), then 

C'(t)=(x′(t),y′(t)) =(0, 0) ↔ α0B(t)=α1D(t) 

=A(t)  tan (
θ

2
) , where A(t) =6t(t-1), B(t)=(3t-1) 

(t-1),  D(t)=t(3t-2). 

We would like to find the MA and MAT for 

the region bounded by the Hermite curve with 

the circular arc from (-cosθ, sinθ) to the point 

(-1,0) rotate about the origin clockwise, we call 

this region R(t;θ,α). 

Theorem 2: Let the Hermite curve 

C(t)=(x(t),y(t))= H1(t;θ,α0,α1), if the local 

maximum curvature is at t0, then 

(x'(t0)y'''(t0)-x'''(t0)y'(t0))(x'(t0)2+y'(t0)2)-3(x'(t0)y

''(t0)-x''(t0)y'(t0))(x'(t0)x''(t0)+y'(t0)y''(t0))=0. 

Notice that this equation has degree 5 in t, 

which means there are 5 local maxi-

mum/minimum curvatures on the curve. 

The MA is very sensitive to its boundary, so 

many researcher use approximated MA instead. 

In pattern recognition, the important information 

for the region is easier to capture by using ap-

proximated MA. Although it could have more 

than one local maximum curvature in our region, 

which means more than one end point of the 

MAT in the region, we would like to find one 

endpoint in the region, so that the MA of the 

region has only two end points (one is on the 

singular point, or the center of the osculating 

circle associated with the boundary curve with 

local maximum curvature, the other one is on the 

origin). In this paper, we use MA to represent the 

approximated MA. 

We start simple case here. Let’s start from 

the case α0=α1=α. Let’s also assume that all 

points on the curve is outside the unit circle and 

the curve has no self-intersection. The MA of a 

region is easy to find if we know the region is 

symmetric to a line, so we have the following 

theorem: 

Theorem 3: The curve C(t)=H1(t;θ,α,α) is 

symmetric to the line y= (−tan 
θ

2
)x. Furthermore, 

C(t) and C(1-t) are symmetric points. 

Proof: Let T = - tan 
θ

2
, we have 

C(t)+C(1−t)

2
=

αTt(t−1)−1

1+T2
.  Obviously, this point is 

on the line y=Tx. 

Notice that this line y=Tx passing through 

C(1/2). Notice further that because C(t) and 

C(1-t) are symmetric points, so the point C(1/2) 

is a singular point, or C(t) is a vertex of the curve, 

which means C(1/2) is a singular point, or has 

constant curvature, or has local maxi-

mum/minimum curvature. 

With the symmetric property, we know the 

MA of the region is on the line (−tan 
θ

2
)x. We 

would like to find the MA point from its foot-

point C(t) or C(1-t), we have the following the-

orem: 

Theorem 4: The medial axis (xm, ym) asso-

ciated with C(t)=(x(t),y(t)), where 0<t<1/2 is (to 

simplify the representation of the equation, we 

eliminate the parameter t for x, y, x′, y′, xm, ym 

and rm):  

(xm, ym) =
(x,y)∙(x′,y′ )

(1,T)∙(x′,y′)
(1,T) and rm = 

|
(1,T)∙(−y,x)

(1,T)∙(x′,y′)
| ‖(x′, y′)‖ 

Where ∙ is the inner product of two vectors, 
|a| is the absolute value of a, and ||(x,y)|| is the 

norm of the vector, T=−tan 
θ

2
. 

With the MA point and its distance to its 

footpoint, we find MAT of the region. We also 

call MAT the medial axis circles in this paper. 

Notice that (xm(0), ym(0)) = (xm(1), ym(1))=(0,0). 

3. Algorithm and Experimental Re-
sult 

In this paper, we want find the medial axis 

circles inside the region, and all medial axis 



Advances in Technology Innovation, vol. 2, no. 1, 2017, pp. 08 - 12 

10 Copyright ©  TAETI 

circles tangent with its neighbors. In our as-

sumption, we can always find one end points of 

the MAT. In some case, there is singular point 

exist, the singular point is the end point of the 

MAT branch (the other one is on the origin). On 

the other case, we can always find a point with 

local maximum curvature; we use the center of 

its osculating circle as the end point of the MAT 

branch. 

How to find normal point of the MAT? 

Consider the end MA point we find, either sin-

gular point or points with local maximum cur-

vature. Assume it has parameter tend, 0<tend<1, 

C(t)=(x(t),y(t)) and we want find t1 and t2, 

0<t0<tend and tend<t1<1, so that C(t1) and C(t2) are 

footpoints of a MA point (x,y), from the prop-

erty of MA point, we have: 

(x-x(t0),y-y(t0))․(x′(t0),y′(t0))=0 

(x-x(t1),y-y(t1)) ․ (x′(t1),y′(t1))=0 

((x-x(t0))
2
+(y-y(t0))

2
=(x-x(t1))

2
+(y-y(t1))

2
 

The first two equations indicates that the line 

through MA point and its footpoints perpendic-

ular to tangent of the footpoint on the curve. The 

third equation indicates that the distances from 

MA point to its two footpoints are equal. 

Now we have 3 equations, and 4 variables, 

which are x, y, t0 and t1. Given t0, we can find the 

MA point (x, y) and t1. The idea to solve the 

system of equation is: 

Algorithm 1: Finding one MA point. 

1. Solve (x,y) by Cramer’s rule from the first 2 

equations, and x, y has two variables t0 and t1. 

2. Substitute the value x, y into the third equations, 

and leave one equation with 2 variables t0 and 

t1. 

3. Given t0, we have one equation with one 

variable t1, so we can find solutions for t1. 

With careful selection of t1, we find the MA 

points with its two footpoints. 

The above idea solves the MAT (MA with 

radius function) for one specified point. As long 

as we know the parameter for the end point, we 

can always find all MAT for parameter from 0 to 

tend.  

We would like to find not only one MA point, 

but also its MA circle tangent to current MA 

circles. We proposed two methods to find next 

MA circle. One method solve the problem al-

gebraically, and the other method using an al-

gorithm with binary search to approach the re-

sult. The first method only suitable for the 

symmetric case, which means the case with 

α0=α1. 

On the case that α0=α1=α, we want find the 

next circle tangent to the first circle, which 

centered at (0,0) with radius 1 in the very be-

ginning. From theorem 4, we know the curve 

whose parameter t associate a MAT circle (xm, 

ym, rm), we assume the next circle whose asso-

ciated boundary parameter is t, so we want to 

find the circle tangent to the unit circle centered 

at the origin. The constraints we give is the dis-

tance between the centers of two circle is equal 

to the sum of the radius of two circles. That is, ‖

(x𝑚, y𝑚)‖ = rm + 1. To find the medial axis circle 

(xm, ym, rm) associated with C(t)=(x(t),y(t)), 

where 0<t<1/2, tangents to the unit circle cen-

tered at the origin, we have to solve a one pa-

rameter equation with degree 10. Because the 

solutions for this equation are sensitive to its 

coefficient, we did not implement this method. 

We use the binary search strategy instead. 

Algorithm 2: Finding the next MA point (eps 

is a small number): 

# Current MA circle is (x,y,r) with parameter t. 

The return value is next MA circle (x1, y1, r1) and 

t1 

1. Set the tmin=t, the current parameter value, and 

tmax is the parameter value for the end MA 

point. 

2. Let dist (distance between two centers) be 2, 

and srad (the sum of the radius of two circles) 

be 1.  

3. While |dist – srad| > eps and |tmax-tmin| < eps : 

tt = (tmin+tmax)/2 

Find the MAT (x1, y1, r1) for the parameter tt 

# (from Algorithm 1). 

Find dist, srad from these two circles. 

If dist - srad > eps, then tmax = tt 

If dist - srad < -eps, then tmin = tt 

4. The value x1, y1, r1 is the MA circle we find. 

From the above algorithm, we can find the 

next MA circles for the parameter t. However, 

when the parameter t value close to the param-

eter associate with the local maximum curvature, 

it can only produce circles intersect with previ-



Advances in Technology Innovation, vol. 2, no. 1, 2017, pp. 08 - 12 

11 Copyright ©  TAETI 

ous circle, which means it cannot find more 

circle tangent to current circle, and "inside" the 

Hermite curve. From the theorem describes in 

the last section, with the idea mentioned above, 

we give 4 examples in this section. The first two 

examples is for the symmetric boundary curve 

where α0=α1=α, for boundary curve with singu-

lar point and with local maximum curvature 

respectively. The third and fourth cases are for 

the non-symmetric boundary curve, also for 

boundary curve with singular point and with 

local maximum curvature respectively. 

Example 1: (Symmetric curve with singular 

point): Given θ=5/9, t=0.5, we can find 

α0=α1=7.15025. The associated MAT is shown 

in Fig. 2. 

 
Fig. 2  Symmetric region with singular point 

Example 2: (Symmetric Curve with local 

maximum curvature): Given θ=5/6, α0=α1=8, the 

associated MAT is shown in Fig. 3. 

 
Fig. 3 Symm. region with localmaximum cur-

vature 

In Fig. 3(a), we find one MA circle tangent 

to the original one. We can find the next circle 

tangent to the red curve and the Hermite cuve, 

however, this circle is not totally “inside” the 

Hermit curve. On this case, we draw the oscu-

lating circle associated with the Hermite curve at 

local maximum curvature (See Figure 3(b)).  

Notice that the above two examples has the 

properties that the MA points are on one straight 

line.The design requirements and design con-

straints are summarized based on the character-

istics of the mechanism. 

Example 3: (Non-symmetric Curve with singu-

lar point): Given θ =5/6, t=0.4, we find 

α0=44.78461, α1=16.79423. The associated 

MAT is shown in Fig. 4. 

 
Fig. 4 Non-Symm. region with Singular point 

Example 4: (Non-symmetric with local maxi-

mum curvature): Given θ=7/6, α0=18, α1=6, the 

associated MAT is shown in Fig. 5. 

 
Fig. 5 Non-Symm. region with maximum 

curvature 

In Fig. 5, the same as Fig 3, the last circle is 

associated with the Hermite curve at local 

maximum curvature. 

4. Conclusions 

The unit circle Hermite curve, H1(t;θ,α0,α1), 

can be used to design curves and images. On 

many cases, the singular point may be needed to 

be considered. For example, the Chinese char-

acters may have cusp on the boundary. Before 

the design of the character, the relationship be-

tween the parameter value and the contour of the 

boundary curve is important. When we give two 

values of the parameters for H1(t;θ,α0,α1) curve, 

the can find the other two values with simple 

computation. When we define a curve, the only 

memory we need is 7 locations to store these 4 

parameters, and other 3 parameters for the 

standardization process. So, the design process 



Advances in Technology Innovation, vol. 2, no. 1, 2017, pp. 08 - 12 

12 Copyright ©  TAETI 

saved not only the time, but also the memory. 

When we find the MA circles inside the region 

bounded by unit circle and H1(t;θ,α0,α1), it is 

possible that the last circle intersects the oscu-

lating circle at the local maximum curvature. 

The osculating circle associated with end MAT 

point of the region, so it is better to show this 

circle, so that the MAT of the region also con-

tains the end MAT point. There are more con-

straints we can used to design the curves. For 

example, the maximum curvature happened at t0, 

the curve passing through a point p0 , the curve 

tangent to a line L0 , and so on. We believe the 

result will as simple as the case we introduced 

here. We can also consider the inverse process, 

that is, from circles tangents to each other, and 

find its boundary Hermite curve. We will leave 

this for further research. 

Acknowledgement 

This work was supported in part by the Na-

tional Science Council in Taiwan under Grants 

MOST 104-2221-E-031-001. 

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