BALKWILL’S ANGLE FOR COMPLETE DENTURES

FINN TENGS CHRISTENSEN, L.D.S.”

S’tavanger, Norway

V ARIATIONS IN THE SIZE of Balkwill’s angle influence the cusp angulation or the sagittal inclination of cuspless posterior teeth in complete dentures. The Balk-

will angle is the inclination of Bonwill’s triangle to the plane of occlusion, measured in the median plane (Fig. 1). According to Balkwilll the size of the angle varies from 22 to 30 degrees, with an average of 26 degrees. This statement was ques- tioned by Kbhler,* who uses a mean value of 21 to 22 degrees. Hart3 reports 20 degrees, and BergstrGm4 found a mean value of 18 degrees roentgenographically with the extremes at 12 and 24 degrees, but with one instance of 8 degrees.

The object of this article is not to discuss the size of Balkwill’s angle but to clarify how variations in it influence the cusp angulation or the inclination of cusp- iess posterior teeth in complete dentures.

INFLUENCE OF BALKWILL’S ANGLE ON THE CHRISTENSEN ANGLE

The calculation of the cusp angulation for complete dentures is based on the Christensen phenomenon and depends on certain suppositions in the calculation of the Christensen angle. 5 It is necessary, therefore, to examine how the size of Balk- will’s angle influences the Christensen angle. The formula for the Christensen angle is:sine[(p +-p) +y] =sine(p+q) +p/ a sine p (Formula II), where p is the inclination of the condylar guidance, cp is the Balkwill angle, y is the Christensen angle, p is the length of protrusion, and a is the height in Bonwill’s triangle.6

The calculation of the Christensen angle in Table I is based on this formula. The angle is calculated for variations in the size of the Balkwill angle as well as in the inclination of the condylar guidance. The protrusion (p) and the height in Bonwill’s triangle (a) remained constant.

According to Table I the Christensen angle (7) increases when the Balkwill angle (g) decreases, except when the condylar guide inclination ‘(p) equals 15 de- grees. The ratio is inverted in this situation. To clarify such an amazing result, the relationship of various Balkwill angles in connection with condylar guide inclina- tions near 15 degrees was investigated. Table II contains a selected range of con- dylar guide inclinations from 5 degrees to 30 degrees, but otherwise it is calculated in the same way as Table I.

A certain conformity is maintained regarding the Christensen angle (7) and the differeltce between the condylar guide inclination and Balkwill’s angle (p +- p)

*Assistant Professor, Norwegian State Dental School, Oslo, Norway.

95

06 CHRISTENSEN J. 1’~s. Um. .Inn.-Frh., 1960

(Table II). This conformity is especially marked when the condylar guide inclina- tion (/3) is equal to 20 degrees and the Balkwill angle is, respectively, equal to 30 degrees and 10 degrees. The Christensen angle is in both cases, the same size (1.15 degrees). The difference between the condylar guide inclination (20 degrees) and the Balkwill angle (10 degrees and 30 degrees, respectively) is in either instance equal to 10 degrees. The other columns show the same relationship; the Christen- sen angle increases when the difference between the condylar guide inclination and the Balkwill angle increases. As a result the cusp angulation, or the inclination of cuspless posterior teeth, varies also because of the relationship between Christensen’s angle and the cusp angulation.

The influence of the Balkwill angle on the Christensen angle is so slight that the alteration of the cusp angulation or the tilting (inclination) of cuspless posterior teeth is insignificant. For example, if the condylar guide inclination is equal to 30 degrees and the Balkwill angle is altered from 30 to 10 degrees, the corresponding

Fig. I.-The Balkwill angle (LAD) is the inclination of Bonwill’s triangle to the plane of occlusion.

difference in the size of the Christensen angle is 0.11 degree with 5 mm. of man- dibular protrusion (Table II). The influence on the cusp height of the difference mentioned is less than 0.1 mm., and the corresponding influence on the inclination of cuspless posterior teeth is less than 1 degree. This is of no importance. With a condylar guide inclination of 20 degrees, even a theoretic difference does not exist in the size of Christensen’s angle when the Balkwill angle is equal to 10 degrees and 30 degrees, respectively. The greatest difference (p = 45 degrees) influences the cusp height by less than 0.2 mm. and affects the inclination of a cuspless second molar by less than 3 degrees. Therefore, the influence of the Balkwill angle on the cusp angulation and on the inclination of cuspless posterior teeth for complete den- tures is not of practical significance.

SIMPLIFIED FORMULA FOR THE CHRISTENSEN ANGLE

Because of the slight influence of the Balkwill angle on the Christensen angle, the formula for the Christensen angle may be simplified. The size of the Balkwill angle is made equal to the inclination of the condylar guidance (p = P) in the

‘I‘.\RI.E: I. THE SIZE OF THE CHRISENSEN ANGLE WHEN THE BALKWILL ANGLE VARIES FKOM 30” ‘ro 10” AND THE CONDYLAR GUDE IXLINATION VARIES FROM 15” ‘ro 45”

I

‘UP I

p“ ~ 30” 26” 22” i 180 14O 10”

‘\ ~ , -----, --____--

19 0.88” 1 0.87” 0.86” 0.86” 0.86” / 0.86” --__ -.-___ _____

30” 1.65” i 1.66” ; 1.68” 1.700 i 1.73” i 1.76” -- .-___

45” I- 2.44” / - ___--.

2.50” 2.57” 2.65” 2.76” 2.91”

(p = Balkaill’s angle 5 = condylar guide inclination Mandibular protrusion = 5 mm

TABLE II. THE SIZE OF THE CHRISTENSEN ANGLE WHEN THE BALKWILL ANGLE VARIES FROM 30” TO 10’ AND THE CONDYLAR GUIDE INCLINATION VARIES FROM 5” TO 30”

-~-___ ~-~_--__ ~ ___~. ~~_ \ I

; )

1 / I \\ cp / I

\ 30” 26” ~ 22” I 18’ s\

\I

/ 1c / 10”

I ..- __- -..- -’ ---- - --._-- ---__-- -~-~-_.~_ _

5” I 0.32” 0.30” 0.30” / 0.29” ( 0.29” -.-__ -

10” 0.61” 0.60” 0.59” 1 0.58” I ~--!?I--

0.57” 0.57”

1.5” i-

0.88 0.87” 1 0.86” 0.86” ( 0.86” I 0.86”

20” 1 1.15” / 1.14” i 1.13 I

1.13” 1.14” 1.15”

25” j 1.40° j 1.40” / 1.40” 1 l.41° 1 1.43” ! 1.45”

30” 1 1.65’

(D = Balkwill’s angle

1.66” 1.68” j 1.70”

6 = condylar guide inclination Mandibular protrusion = 5 mm.

TABLE III. THE SIZE OF THE CHRISTENSEN ANGLE, ACCORDING TO BOTH FORMULA II AND FORMULA VI, WHEN THE INCLINATION OF THE CONDYLAR GUIDANCE VARIES FROM 10’ TO 50”

-.~ y according to Formula II -____ y according to Formula VI

5 = condylar guide inclination 7 = Christensen’s angle Mandibular protrusion = 5 mm.

98 CHRISTENSEN J. Pros. Den. Jan.-Feb., 1960

formula for the Christensen angle. The simplified formula may be expressed: sine y = p/a sine /3 (Formula VI), where y is the Christensen angle, /3 is the in- clination of condylar guidance, p is the length of protrusion, and a is the height in Bonwill’s triangle.

The difference between the exact value of the Christensen angle, according to Formula II, and the approximate value, according to Formula VI, is given in Table III. The difference is insignificant.

SUMMARY

The effect of the size of Balkwill’s angle on the Christensen angle and on cusp angulation was discussed, and on the basis of the findings a simplified formula for the Christensen angle was suggested.

REFERENCES

1. Balkwill, F. H.: The Best Form and Arrangement of Artificial Teeth for Mastication, Brit. J. D. SC. 9:278-282, 1886.

2. Kohler, L. : Die Vollporthese, in Scheff and Pichler, editors : Handbuch der Zahnheilkunde, Band IV, Berlin and Wien, 1929, Urban & Schwarzenberg, p. 286.

3. Hart, F. L. : Full Denture Construction, J.A.D.A. 26:455-461, 1939. 4. Bergstrom, G.: On the Reproduction of Dental Articulation by Means of Artciulators, Acta

Odont. Scandinavica 9: suppl. 4, p. 49, 1950. 5. Christensen, F. T.: Cusp Angulation for Complete Dentures, J. PROS. DEN. 8:910-923, 1958.

KANNIKGATEN 13 STAYANGER, NORWAY