THE EFFECT OF INCISAL GUl&MNCE ON CUSP ANGULA’MON IN FROSTHE’RC QCCLTJSIQN

FINN TENGS CHRISTENSEN, L.D.S.” Stavanger, Norway

T HE OBJECT of this article is to clarify the relationship between the sagittal cusp angulation and the inclination of the incisal guidance. Swenson1 has expressed

the relation by means of the following formula:

cusp lncisal inclination = inclination

+ Fraction of distance Condyle In&al from incisal g&&me inclination angle - inclination angle >

If the cusp angulation is 8, the inclination of incisal guidance is U, the condplar guide inclination is /3, and the fraction of the distance from the incisal guidance is ri. Swenson’s formula may be expressed : 2 = u + d (@ - U) .

Therefore, if the cusp angulation is to be in accordance with the incisal guid- ance and the condylar guidance, the cusp angulation varies as follows :

1. If the incisal inclination is less than the inclination of condylar guidance, the cusp angulation must equal the incisal inclination plus an angle that depends on the difference p - u and the fraction of the distance from the incisal point to the cusp in question.

2. If the inclination of the incisal guidance is equal to the condylar guidance, that is, u = p, the cusp angulation is equal to both the inclination of incisal guid- ance and the inclination of the condylar guidance, that is, 2 = u = fi (Fig. 1).

3. If the incisal guide inclination is greater than the condylar guide inclination, the difference p - u will be negative. The cusp angulation is equal to the incisal guide inclination minus an angle that depends on the difference p - u and the frac- tion of the distance of the cusp in question from the incisal point.

THE DISTANCE FROM TKE INCISAL POINT

Swenson’s formula is an empirical formula and logically unassailable. How- ever, it is not an exact formula. The factor d, the fraction of distance from the incisal point to the cusp in question, is the weak link. The problem is to calculate d. To solve that problem, it is necessary to determine the cusp angulation for complete dentures and the suppositions regarding the calculations of the cusp angulation when the incisal guidance is zero degrees.2

The calculation of the cusp angulation for complete dentures is based on the calculation of the Christensen angle (y), The protrusive movement of the condylar

*Asefstant Professor, Norwegian State Dental School, Oslo.

48

;;hltlp,“r 11’ EFFECT OF INCISAL GUIDANCE ON CUSP ANGULATION 49

axis is divided into two separate movements. 2 The first movement is a parallel shift of the incisal point and the condylar axis along the incisal guidance which is as- sumed to be zero degrees.

The same principle is used to clarify the influence of the incisal guidance on the cusp angulation. In the present situation, the first movement is a parallel shift of, respectively, the condylar axis and the incisal point along the incisal guidance. According to Fig. 2, the condylar axis (L) is thought to move along an imaginary condylar path (LL,) which is situated parallel to the incisal guidance (AA,). The mandibular occlusal plane (AK) is shifted parallelly to a new position (A,K,). The next step is a rotation of the mandible around the incisal point (A,) in the new position, until the condylar axis L, touches the real condylar path. L, shifts to L,

Fig. I.-The inclination of the in&al guidance is equal to the incl>nation of the condylar guidance. The incisal point and the projection of the condylar axis move on parallel paths. To maintain antagonistic tooth contact along the entire length of the protrusive facets, the sagittal cusp angulation of all of the cusps must be the same size as the inclination of the incisal and the condylar guidances.

if the condylar guide inclination is greater than the incisal guide inclination, If the relation is inversed, that is, the incisal guide inclinaton is steeper than the condylar guide inclination, L, shifts to L,.

The angle K,A,K2 is formed if the condylar path inclination is steeper than the incisal guide inclination. If the relation is inversed, the angle K,A,K, is formed. These angles correspond to the Christensen angle, and the size depends upon the difference between the condylar guide inclination and the incisal guide inclina- tion.

If the condylar guide inclination is the same size as the incisal guide inclina- tion, the cusp angulation must be of the same size (/3 = I) = 2) because the incisal point and the condylar axis move on parallel paths (Fig. 1) . That is in accordance with Swenson’s formula. However, when there is any difference between the incisal and the condylar guide inclinations, the cusp angulation must be altered to attain complete antagonistic contact along the entire length of the protrusive facets during protrusive movements.

50 CHRlSTENSEIi J. P:ua. !>tn. Jan.-F.& 1961

The difference between the incisal and cmdylar angles (p -- V) is called the cid angle (condylar-incisal difference angle ) If the inclination of condylar guidauce is greater than the inclination of incisal guidance. the cusp nngulation is equai to

Fig. Z.-The inclination of the incisal guidance is different from the inclination of the condylar guidance: (A) the incisal point in centric occlusion; IA,) the position of the in&al point after a protrusive movement along the incisal path; (AA,) the length of the protrusion; (AK) the plane of occlusion; (A,K,) an auxiliary line indicating the mandibular occlusal plane after parallel shifting of the condylar axis and the incisal point along the in&al guidance; (A&e) the mandibular occlusal plane after a protrusive gliding movement of the in&al point along the incised path and of the condylar axis along the condylar path.

(J&j Auxiliary letter; (L) the position of the candylar axis in centric occlusion; (L1) tbcs position of the condylar axis after a protrusive movement along an imaginary path (LLJ parallef to the incisal guidance (AAJ; c.52) the condylar axis after a protrusive movement on the con- dylar path (LLJ; (L$ the position of the condylar axis after a protrusive movement if the inclination of condylar guidance is less than the inclination of the in&al guidance.

(C) The molar point in the centric occlusion; fC,j the molar point after a parallel shifting of the in&al point and the condylar axis on, respectively, the incisal path and an imaginary condylar path (.Ud parallel to this; (Cd the position Of the molar point after a protrusive move- ment with the incisal point along the ineisal path (AA,) and the condylar axis along the condylar path (LLd.

The angles y and r, correspond to the Christensen angle for a condylar guide inclination equal to &LL,. LILLa is equal to the c&l angle, WhtCh is the di%rence between the condylar guide in&nation and the in&al guide inelfira$&n (,T -a?. Consequently, the cusp angulation equ&s the incisal guide inclination (see Fig. 1) with a correction depending on the cld angle. Each angle of condylar guitianw, as well as each CM angle, has a corresponding Christensen angle, T&e calculation of the correction is based on the ChPfstensen angle and can be carried out in Table I. The result is shown in Table II.

(Caj The position of the molar point after a protrusive movement, when the inclination of the in&al guidance is steeper than the inclination of the condylar guidance.

Volume 11 Number 1 EFFECT OF INCISAL GUIDANCE ON CUSI? ANGULATION 51

u plus an angle that depends upon the cid angle (p - u). If the relation is inversed (/3 is less than u), the cusp angulation is equal to v minus an angle depending on the cid angle.

The influence of the cid angle on the cusp angulation when v equals zero de- grees is expressed in Table I, where the cid angle is equal to the inclination of condylar guidance because p - v = p - 0” = /?. Table II is in principle the same as the Cusp Incline Table, but the inclination of the condylar guidance has been replaced by the cid angle.

TABLE I. CUSP INCLINE TABLE

I CONDYLAR GUIDE ANGLE ( 0)

I______ -___- _____

2% 5” 10” 15” 20” 25” 3o”

ZMt 4O 8” 12” 16 20” 24”

EM1 3” 6” 9O 12” 15” 18”

ZPlPZ 2” 4” 6’ 8” 10” 12”

-___

This table indicates the relation between the sagittal cusp angle (or the tilting of cuspless posterior teeth) (r )and the inclination of condylar guidance I,¶).

M = molar; P = premolar.

Table I is based on the following ratio between the cusp angulation and the inclination of condylar guidance2 :

ZM, = 5/1OB

EM* = 4/roa

ZMl = s/lop ZP,P, = z/lop

Expressed by Swenson’s formula, 2 = v + d (/? - u), the relation (with u=O’) is:

zM, = 0” + S/10@ - on>

2% = 0” ‘+ 4/10@ - 00) zM, = 0” + 3/10(8 - 0’) ZP,P, = 0” + 2/1O(j3 - 0”)

With an incisal guide inclination equal to U, t.he relation is :

ZM, =v + S/10@ - u) zM, =v + 4/10(/3 - u) EM, = v + 3/10@ - v) EP,P, = ” + 2/1O(gl - u)

The factor d is consequently S/l0 for the third molar, 4/10 for the second molar, 3/10 for the first molar, and Z/10 for the premolars.

TABL

E II.

C

ID

INC

LIN

E TA

BLE

~.-.-

--___

--_I_

--.

- ~~

. I_-~

~.

..-.

____

CU

SP A

NG

LE (

I;)

CID

AN

GLE

(fi

- LJ

) __

_~-_

I_.

_ ---

I- __

--.---

g

BM,

(d =

S/

lo)

E

ZM,(d

=4/1

0)

I %

lJ

+ 12

” (

u-j-

8”

1 vt

- 4’

g

ZM,

(d =

3/

10)

/ R

z

ZF’lP

z (d

=

2/10

) -_

_ .-I

___-

~_

..~

---

..__

__---

.- . .

.~_~

____

-._

.._

_ _

.__.

Th

is

tabl

e in

dica

tes

the

rela

tion

betw

een

the

sagi

ttal

cusp

an

gle

(or

tiltin

g of

cu

sple

ss

post

erio

r te

eth)

(2

: )

and,

re

spec

tivel

y,

the

cond

ylar

(1

9) a

nd

the

inci

sal

(u)

guid

e in

clin

atio

ns.

The

calc

ulat

ion

of

cusp

an

gles

is

ba

sed

on

the

CM

an

gle,

w

hich

is

th

e di

ffere

nce

betw

een

the

cond

ylar

an

d th

e in

cisa

l gu

ide

incl

inat

ions

(,3

-

v)

and

the

fract

fon

of

the

dist

ance

(d

) fro

m

the

in&a

l po

int

to

the

cusp

in

qu

estio

n.

M =

m

olar

; P

= pr

emol

ar.

x:%4 ‘1’ EFFECT OF INCISAL GUIDANCE ON CUSP ANGULATION 53

The formula may also be expressed 9 = II + ‘E cid, where cid is equal to the angle p -- V. Table II is based on this formula. For example, an incisal guide in- clination of 10 degrees and a condylar guide inclination of 40 degrees give the fol- lowing cusp angulations :

ZM, = v + d(p - u), where d for M, = S/10 ZM, = 10” + S/10(40” - 100) = 10” + 15” = 25”

In the same way, the angulation for the rest of the cusps may be calculated :

EM, = 10” + 1-p = 22” ZMl = 10” + 9” = ‘19” zp,p* = 10” + 6” = 16”

If the incisal guide inclination is equal to 40 degrees and the condylar guide inclination is equal to 10 degrees, the cusp angulation for Ma is:

zM, = 40” -I- 5/10(10° - 40”) = 40” + 5/10(--30”) = 40” - 15” = 25”

In the same way, the angulation for the rest of the cusps may be calculated :

XMZ = 40” - 12” = 28”

-,f, = 40” - 9” = 31”

zp,p* zz Jo’ - (j” = 34”

The values in the tables are approximate. The difference between the approxi- mate values and the exact values is less than 1 degree for cid angles up to 60 degrees.2

GYSI’S FORMULA FOR THE SECOND MOLAR

Gysi’s formula for the angulation of the second molar3 is :

Angulation of M, = Condyle guide inclination f Incisal guide inclination

2

If the angulation of M, is SM,, the condylar guide inclination is p, and the p + u.

incisal guide inclination is V, the formula will be 8M, = 2 .

Gysi’s formula for M, is in accordance with Swenson’s formula for Ma, where : 13+u

zM, = u -t d(a - u) = u C S/10@ - v) = $ + l/2(@ - u) = 2

Gysi’s formula is assumed to be an empirical formula, giving the dentist a conception of the approximate cusp angulation and the setting of the incisal table of an ad, justable articulator.

INFLUENCE OF THE COMPENSATING CURVE OF OCCLUSION

Regarding the culculation of the cusp angle in Table II, the summits of the cusps are assumed to be situated in the plane of occlusion. By means of the compen- sating curve of occlusion, the cusp angulation may be reduced and still maintain antagonistic contact along the entire length of the facets by protrusive gliding

54 CHRISTENSEN J. Pros. Uen. Jan..Feb., 1961

movements. Taking advantage of the compensating curve, the protrusive facets ought to be reduced correspondingly to the cusp plane angle, which is the sagittal slope of the tooth to the occlusal plane.4

Setting cuspless posterior teeth (cusp angulation equals zero degrees) in COIII- plete dentures, where the incisal guidance equals zero degrees, the cusp pfanr angle equals the slope of the cusp angulation according to Table I.4 The same line of argument is applicable to positive incisal guide inclination. To attain :L smooth-running antagonistic contact along the entire length of the protrusive facets by gliding movements of the jaw, the slope of the cuspless posterior teeth may be set in accordance with the cusp angulation in Table TT.

SUMMARY AND CONCLUSION

The relation between the incisal guide inclination (u) , the cusp angulation (2). and the condylar guide inclination (p) can be expressed by the formula B = 11 + d (p - u), where d (the fraction of distance from the incisal point) has values for the respective cusps of M, = 5/10, M, = 4/10, M, = 3/10, and P,P, = 2110. The exactness of the formula is approximately 4 1 degree for differences between the condylar guide inclination and the incisal guide inclination until a difference (cid angle) of 60 degrees is reached. To attain balanced occlusion by using cuspless posterior teeth, it is necessary to regard the entire occlusal surface of the cuspless tooth as one single protrusion facet and set up the teeth accordingly.

REFERENCES

1. Swenson, M. G.: Complete Dentures, ed. 2, St. Louis, 1947, The C. V. Mosby Company.

2. Christe&Zn?9fi T Cusp Angulation for Complete Dentures J PROS DEN. 8:91Q-923 1958. 3. Gysi, A.: kieferbkwegung und Zahnform, in Scheff and i>idhler, editors: Handbdh der

Zahnheilkunde, Berlin and vienna, 1929, Urban and Schwarzenberg, vol. IV, p. 127. 4. Christey;? l&I’. : The Compensatmg Curve for Complete Dentures, J. PROS. DEN. 10:637-

9 . KANNIKGATEN 13 STAVANCER, NORWAY