Describing the geometric difference of architectural forms in three primary shapes of circle, triangle and square

Figures & data

Figure 1. Façade shapes of Ronchamp Chapel and Guggenheim Museum Bilbao

Table 1. Measurements of circularity

Label	Expression
C1	C1 = $\frac{4\pi A}{L^2}$, where A = Area of the object polygon, L = Perimeter of the object polygon
C2	C2 = C1CH·C, where C = A/ACH, CH = Convex Hull of the polygon, C = Convexity
C3	C3 = 1 – (σrR/RRCA) ri = radial distance from centre to each vertex, RRCA = radius of RCA
C4	C4 = 1 – Δ(PRCA, Po), Δ(PRCA, Po) = (${\mathrm{P}}_{\mathrm{R}\mathrm{C}\mathrm{A}}\oplus {\mathrm{P}}_{\mathrm{o}}$)/(PRCA ∪ Po)
C5	C5 = 1 – $({\sigma }_{\hat{r}R}/R_{RCR}),{\hat{r}}_i$= radial distance from the center to evenly distributed points on the boundary line. RRCR: radius of the RCR
C6	C6 = 1- Δ(PRCR, Po), Δ(PRCR, Po) = (${\mathrm{P}}_{\mathrm{R}\mathrm{C}\mathrm{R}}\oplus {\mathrm{P}}_{\mathrm{o}}$)/(PRCR ∪ Po)
C7	C7 = RMIC/RMCC
C8	C8 = AMIC/A
C9	C9 = AMIC/AMCC
C10	C10 = A/AMCC
C11	C11 = $1-\frac{{\sigma }_{\varphi \pi }}{\pi }$, ϕ = mod(θ, π)
C12	$\psi \cdot r$, for |ψ| < 1.0, where ψ = θ/π, r = segment length(|ψ| <1.0)/total perimeter
C13	sin(θ/2), for |ψ| < 1.0, r = segment length(|ψ| <1.0)/total perimeter

Table 2. Measurements of triangularity

Label	Expression
T1	T1 = $\frac{12A\sqrt{3}}{L^2}$ A = Area of the P, L= Perimeter of the object polygon, P = object polygon of the building shape.
T2	T2 = ART/A, ART = Area of RT derived from P, RT = (p1, p2, p3), where $p_1=\left(\left(x\right),\left(y\right)\right),p_2=\left(\left(x\right),\left(y\right)\right),p_3=\left(x\left(\left(y\right)\right),max\left(y\right)\right)$
T3	T3 = T2∙ T1RT, T1RT: T1 value of RT derived from P.
T4	T4 = T1RT ∙ Δ(PRT, P), Δ(PRT, P) = (PRT $\oplus$ P)/(PRT ∪ P)
T5	T5 = A/ARTE, A = Area of P, ARTE = Area of PRTE, RTE = (p1, p2, p3), where $p_1=\left(\left(x\right),\left(y\right)\right),p_2=\left(\left(x\right),\left(y\right)\right),p_3=\left(\frac{\left(x\right)+\left(x\right)}{2},\left(y\right)+h\right)$
T6	T6 = 1- Δ(RTE, P), Δ(RTE, P) = A(RTE $\oplus$ P) /A (RTE ∪ P)
T7	T7 = 1 – Δ(RTA, P), Δ(RTA, P) = A(RTA $\oplus$ P)/A(RTA ∪ P)
T8	T8 = T1CH∙ convexity, CH = convex hull of P, convexity = ACH/AP
T9	T9 = α∙Δ(Ps, P), α = $sin\left(\frac{3}{2}\theta \right)for0\le \theta \le \frac{\pi }{3}$, $sin\left(\frac{3}{2}\theta +\pi \right)for-\frac{\pi }{3}\le \theta <0$, $sin\left(3\theta -\frac{\pi }{2}\right)for\frac{\pi }{3}\le \theta <\frac{\pi }{2},$ $sin\left(3\theta +\frac{3\pi }{2}\right)for-\frac{\pi }{2}\le \theta <-\frac{\pi }{3}$, $sin\left(\frac{\pi }{2}-\theta \right)for\frac{\pi }{2}\le \theta <\pi ,-\pi \le \theta <-\frac{\pi }{2}$, Ps = simplified polygon of P
T10	T10 = α∙Δ(Ps, P)∙ symm(sl, sr), symm(sl, sr): the ratio between the projected lines of the left side(sl) and right side(sr) to the base.
T11	T11 = β∙Δ(Ps, P) $\beta$: the difference between the lengths of the three sides

Table 3. Measurements of squareness

Label	Expression
S1	S1 = 16A/L^2 if 16A<L^2, L^2/16A if 16A > L^2 A = Area of the object polygon, L= Perimeter of the object polygon
S2	S2 = S1CH ∙ C, C = A/ACH, (CH: the convex hull of the polygon, C: Convexity)
S3	S3 = R1∙ AR, R1 = A/AAABB, (AR: aspect ratio = min(width, height)/max(width, height)
S4	S4 = (S4x + S4y)/2, wherer S4x = $1-\frac{{\sigma }_{lx}}{SL}$, S4y = $1-\frac{{\sigma }_{ly}}{SL}$, SL = Side Length of RSA (√A), ${\sigma }_{lx}=\sqrt{\frac{1}{n}\sum _{i=1}^n{\left(l_{ix}-SL\right)}^2},{\sigma }_{ly}=\sqrt{1\sum _{j=1}^m{\left(l_{jy}-SL\right)}^2}$, lx = horizontal length, ly = vertical length
S5	S5 = (S5x + S5y)/2, S5x = $1-\frac{{\sigma }_{lx}}{M}$, S5y = $1-\frac{{\sigma }_{ly}}{M}$, M = $\frac{\overline{l_x}+\overline{l_y}}{2}$, $\overline{l_x}$, $\overline{l_y}$: mean of x-axis and y-axis length, ${\sigma }_{lx}=\sqrt{\frac{1}{n}\sum _{i=1}^n{\left(l_{ix}-M\right)}^2},{\sigma }_{ly}=\sqrt{1\sum _{j=1}^m{\left(l_{jy}-\mathrm{M}\right)}^2}$
S6	S6 = 1- Δ(PRSA, Po), Δ(PRSA, Po) = (PRSA $\oplus$ Po)/(PRSA ∪ Po)
S7	S7 = 1 – Δ(PRSD, Po), Δ(PRSD, Po) = (PRSD $\oplus$ Po)/(PRSD ∪ Po)
S8	R2∙AR2, AR2 = aspect ratio of RRD, (R2 is defined below)
S9	R3∙AR3, AR3 = aspect ratio of R3
S10	R3CH∙AR3∙convexity
R2	R2 = (R2x + R2y)/2, R2x = $1-\frac{{\sigma }_{lx}}{\overline{l_x}}$, R2y = $1-\frac{{\sigma }_{lx}}{\overline{l_y}}$
R3	$1-\frac{{\sigma }_{\varphi ,\pi /2}}{\pi /2}$, θ: internal angle of the simplified polygon, ϕ: θ mod π

Figure 2. Line simplification

Figure 3. Examples of polygon approximation

Figure 4. Boolean operations of two overlaid polygon

Figure 5. Reference circles with variant shapes for the circularity measurements

Figure 6. (a) RT and (b) RTE of four building facades. (c) RTE and (d) RTA of twelve geometries. The shaded region indicates the discrepancy area while the region filled with solid black indicates the overlapping area

Figure 7. Reference shapes for the squareness measurements; CH, AABB, RSA, RSD, RRD marked in different line styles

Figure 8. The process of building RSD

Figure 9. The regions of the exclusive area between RSA and P on the left and RSD and P on the right. The shaded region indicates discrepancy while the solid black indicates the overlapping area

Figure 10. Deformations of a reference shape at regular intervals and their correlations with the measures of circularity

The numeric value e, δ, α indicates the degree of deformation estimated independently according to the method of deformation. (* Circularity methods with the highest correlation)

Figure 11. Deformations of a reference shape at regular intervals and their correlations with the measures of triangularity

Figure 12. Deformations of a reference shape at regular intervals and their correlations with the measures of squareness

Table 4. Correlation coefficients between the measurement methods of circularity and the degrees of deformation with known intervals in a circle

Type	A	B	C	D	E	F	G	H	I	J	Mean R^2
df	32	32	33	32	32	64	60	33	60	60
C1	0.9465	0.8707	0.9166	0.9736	0.9860	0.9345	0.9741	0.9812	0.9867	0.9761	0.9125
C2	0.9465	0.9992	0.9697	1.0000	0.9860	0.9998	0.9953	0.9812	0.9867	0.9993	0.9732
C3	0.9735	0.8284	0.7672	0.9925	0.9261	0.9795	0.9778	0.9938	0.9545	0.9887	0.8857
C4	0.9996	0.9837	0.9953	0.9925	0.9977	0.9859	0.9978	0.9802	0.9951	0.9995	0.9856
C5	0.9990	0.9983	0.9511	0.9908	0.9997	0.9844	0.9946	0.9926	0.9973	0.9986	0.9815
C6	0.9986	0.9769	0.9813	0.9943	0.9984	0.9882	0.9948	0.9831	0.9954	0.9997	0.9823
C7	0.9996	0.9978	0.9992	0.9611	0.9933	0.9855	0.9938	0.9958	0.9955	0.9988	0.9842
C8	0.9995	0.9760	0.9788	0.8147	0.9768	0.9384	1.0000	0.9540	0.9864	0.9954	0.9282
C9	0.9843	0.9572	0.9727	0.8926	0.9483	0.9378	0.9624	0.9907	0.9651	0.9838	0.9214
C10	0.9996	0.9793	0.9951	1.0000	1.0000	1.0000	0.9766	0.9995	0.9986	1.0000	0.9898^(2^)
C11	−0.9382	0.9116	0.9318	0.9059	0.9979	0.9530	0.8941	−0.9970	0.9942	0.9173	0.8927
C12	0.0722^(1)	0.4085	0.9294	0.9592	0.9576	0.7954	0.9463	0.9961	0.9112	0.9149	0.7060
C13	0.8963	0.4095	0.9145	0.9583	0.9575	0.7955	0.9413	0.9961	0.9225	0.9129	0.7838

⁽¹⁾All others are significant at p < 0.01 except this.

⁽²⁾Highest mean of R² (mean R = .9949)

Table 5. Correlation coefficients between the measurement methods of circularity and the degrees of deformation with known intervals from an equilateral triangle

Type	A	B	C	D	E	F	G	H	I	J	Mean R^2
df	32	32	32	32	32	32	32	32	32	32
T1	0.9542	0.5592	0.8481	0.9812	0.7926	0.9888	0.9805	0.9143	0.9897	0.4075	0.7454
T2	−0.1814	1.0000	0.9056	0.2706*	1.0000	0.8032	0.9776	0.8452	0.9989	0.9995	0.7238
T3	0.9542	0.9706	0.9804	0.9808	1.0000	0.9505	0.9697	0.8664	0.9989	0.9999	0.9368
T4	0.9542	0.9706	0.9804	0.9808	1.0000	1.0000	0.9697	0.9568	0.9986	0.9985	0.9626
T5	1.0000	1.0000	1.0000	0.9925	1.0000	0.5332	0.9912	0.7122	0.9933	0.9998	0.8745
T6	1.0000	1.0000	1.0000	0.9963	1.0000	0.9955	0.9912	0.9752	0.9977	0.9974	0.9907
T7	0.9949	0.9414	0.9751	0.9906	0.9166	0.9979	0.9893	0.9336	0.9886	0.9977	0.9467
T8	0.9542	0.5592	0.8481	0.9812	0.5106	0.9527	0.9805	0.9873	0.9897	0.9837	0.7957
T9	0.9378	0.9767	0.9958	0.9870	0.9806	0.5162	0.9368	0.9650	0.9800	−0.0749	0.7802
T10	0.9378	0.9767	0.9958	0.9978	0.9806	0.7469	0.9576	0.9197	0.9800	−0.0749	0.8069
GT	0.9542	0.8479	0.9940	0.9816	−0.0118*	0.9788	0.8676	0.9188	0.8111	0.9404	0.7678

1) correlation is significant at the 0.01 level except marked with *

Table 6. Correlation coefficients between the measurement methods of circularity and the degrees of deformation with known intervals from a square

Type	A	B	C	D	E	F	G	H	I	J	Mean R^2
df	32	32	32	32	32	32	32	32	32	32
S1	0.9202	0.9597	1.0000	1.0000	0.9267	0.9851	0.9653	0.9895	0.9141	0.9986	0.9341
S2	0.9202	0.9597	0.9923	0.9922	0.8714	0.9759	0.9715	0.9952	0.9070	0.9982	0.9202
S3	1.0000	1.0000	1.0000	1.0000	0.8319	0.9978	0.9993	0.9973	0.9998	0.9911	0.9663
S4	0.9843	0.9962	0.9066	0.8849	0.9636	0.9914	0.9851	0.9992	0.9898	0.9967	0.9419
S5	0.9843	0.9955	0.9256	0.9245	0.9796	0.9900	0.9849	0.9993	0.9893	0.9948	0.9548
S6	0.9981	0.9733	0.9941	0.9962	0.9863	0.9987	0.9996	0.9952	0.9993	0.9698	0.9823
S7	0.9968	0.9937	0.9822	0.9852	0.9860	0.9988	0.9999	0.9954	0.9991	0.9986	0.9872
S8	1.0000	0.9932	0.9256	0.9559	0.7974	0.9956	0.9996	0.9921	0.9989	0.9862	0.9338
S9	1.0000	0.9995	0.9371	0.9885	0.7525	0.9282	0.9934	0.9884	0.9932	0.9779	0.9189
S10	1.0000	0.9946	0.9404	0.8714	0.7119	0.6711	0.4812	0.5291	0.6976	0.9184	0.6432
R1	−0.0234*	1.0000	1.0000	1.0000	0.8015	0.9995	0.9246	0.9921	0.8082	0.9939	0.9024
R2	0.2575*	0.9955	0.9256	0.9249	0.7879	0.9871	0.9303	0.9918	0.8396	0.9892	0.8701
R3	NA	0.9995*	0.9371	0.9885	0.2072	0.9309	0.7829	0.9897	0.5553	0.9503	0.6961

* all r values except the three marked are significant at p > 0.01, and the values that are not significant at p < 0.01 are excluded from averaging the R² measure.

Table 7. Mean of R squared between the measurement methods and the degrees of deformation

C	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10	C11	C12	C13
mean R^2	0.9125	0.9732	0.8857	0.9856	0.9815	0.9823	0.9842	0.9282	0.9214	0.9898	0.8927	0.706	0.7838
T	T1	T2	T3	T4	T5	T6	T7	T8	T9	T10	GT
mean R^2	0.7454	0.7238	0.9368	0.9626	0.8745	0.9907	0.9467	0.7957	0.7802	0.8069	0.7678
S	S1	S2	S3	S4	S5	S6	S7	S8	S9	S10	R1	R2	R3
mean R^2	0.9341	0.9202	0.9663	0.9419	0.9548	0.9823	0.9872	0.9338	0.9189	0.6432	0.9024	0.8701	0.6961

Table 8. Shape measurements of 30 buildings classified into two types of shapes

Building	Facade Floor pl.	Circularity	Equilateral triangularity	Squareness	Deformity	Dominant shape	Dominance	Percentage difference with mean dominance
Ronchamp Chapel	Facade	0.5334	0.7130	0.5526	0.4003	ET	0.1133	19.43
	Floor Pl.	0.4911	0.5723	0.6477	0.4296	S	0.0773	21.39
Sydney Opera House	Facade	0.2151	0.3760	0.3981	0.6703	S	0.0684	30.73
	Floor Pl.	0.5452	0.5693	0.5890	0.4322	S	0.0212	93.67
Villa Rotonda, Vicenza	Facade	0.3590	0.6293	0.5247	0.4957	ET	0.1250	29.11
	Floor Pl.	0.7917	0.4400	0.6958	0.3575	C	0.1492	43.60
Temple of Khandariya Mahadeo, Khajuraho	Facade	0.3717	0.5784	0.4575	0.5308	ET	0.1092	15.77
	Floor Pl.	0.3671	0.4866	0.5350	0.5371	S	0.0721	26.78
Taj Mahal, Agra	Facade	0.4639	0.6362	0.5345	0.4551	ET	0.0913	2.10
	Floor Pl.	0.7811	0.4356	0.9326	0.2836	S	0.2162	121.27
Villa Savoye, Dessau	Facade	0.4290	0.5776	0.4678	0.5085	ET	0.0861	7.96
	Floor Pl.	0.7399	0.5533	0.8296	0.2924	S	0.1220	23.49
Tokyo Olympic Hall, Tokyo	Facade	0.1850	0.3139	0.3793	0.7073	S	0.0866	7.38
	Floor Pl.	0.3146	0.5352	0.5578	0.5308	S	0.0886	12.35
North Christian Church	Facade	0.1715	0.4743	0.1994	0.7183	ET	0.1926	69.52
	Floor Pl.	0.6537	0.6019	0.6836	0.3536	S	0.0372	42.99
Guggenheim Museum, Bilbao	Facade	0.1406	0.2469	0.2532	0.7864	S	0.0396	80.76
	Floor Pl.	0.2903	0.4575	0.3359	0.6388	ET	0.0963	5.48
Mt. Angel Abbey Library, St. Benedict	Facade	0.2497	0.3506	0.4686	0.6437	S	0.1123	18.55
	Floor Pl.	0.4312	0.6649	0.6161	0.4293	ET	0.0942	5.58
Pantheon, Rome	Facade	0.6938	0.6204	0.7615	0.3081	S	0.0696	29.03
	Floor Pl.	0.5176	0.5766	0.6323	0.4245	S	0.0568	51.05
Mingtang-Biyong Ritual Complex, Xian	Facade	0.3925	0.6799	0.5633	0.4548	ET	0.1347	36.38
	Floor Pl.	0.8108	0.4400	0.7781	0.3237	C	0.1345	29.28
St. Michael, Hildesheim	Facade	0.3788	0.5621	0.4286	0.5435	ET	0.1056	12.44
	Floor Pl.	0.4367	0.5014	0.5062	0.5186	S	0.0248	73.23
Great Stupa, Sanchi	Facade	0.2697	0.4837	0.5348	0.5706	S	0.1054	12.25
	Floor Pl.	0.7461	0.4557	0.5977	0.4002	C	0.1463	44.93
Philharmonic Hall, Berlin	Facade	0.3675	0.6092	0.5171	0.5021	ET	0.1113	17.66
	Floor Pl.	0.6072	0.4926	0.7852	0.3717	S	0.1569	53.70
Price Tower, Oklahoma	Facade	0.3250	0.1587	0.3810	0.7118	S	0.0928	0.47
	Floor Pl.	0.6624	0.5714	0.6881	0.3594	S	0.0475	54.56
Blues Point Tower, Sydney	Facade	0.3279	0.1299	0.3705	0.7239	S	0.0944	1.24
	Floor Pl.	0.6339	0.4747	0.9139	0.3258	S	0.2397	143.50
Nexus World Housing (Steven Hall), Fukuoka	Facade	0.2859	0.4441	0.3140	0.6520	ET	0.0961	3.02
	Floor Pl.	0.2547	0.3690	0.2979	0.6928	ET	0.0618	39.06
The Whale, Amsterdam	Facade	0.3103	0.4514	0.3442	0.6314	ET	0.0828	11.86
	Floor Pl.	0.3365	0.2755	0.2984	0.6965	C	0.0330	73.39
Leicester Univ. Eng. Bldg., Leicester	Facade	0.4708	0.7255	0.4998	0.4346	ET	0.1601	52.79
	Floor Pl.	0.6315	0.6326	0.6324	0.3678	ET	0.0004	76.65
Arab World Institute, Paris	Facade	0.4672	0.6014	0.4911	0.4801	ET	0.0815	13.43
	Floor Pl.	0.3248	0.4790	0.4121	0.5947	ET	0.0737	29.10
London City Hall, London	Facade	0.5069	0.6243	0.6900	0.3929	S	0.0829	11.74
	Floor Pl.	0.9974	0.4337	0.7367	0.2774	C	0.2748	190.87
Mercedes-Benz Museum, Stuttgart	Facade	0.5485	0.6230	0.6499	0.3929	S	0.0428	74.15
	Floor Pl.	0.7267	0.5789	0.6675	0.3423	C	0.0690	43.56
30 St. Mary Axe, London	Facade	0.3196	0.1604	0.4172	0.7009	S	0.1181	23.53
	Floor Pl.	0.9927	0.4360	0.7393	0.2773	C	0.2700	155.69
Seattle Library, Seattle	Facade	0.5260	0.6958	0.7770	0.3337	S	0.1107	17.13
	Floor Pl.	0.6322	0.4873	0.8916	0.3296	S	0.2212	114.86
Hu-Tong House (Waro Kishi), Japan	Facade	0.2774	0.3959	0.3194	0.6691	ET	0.0650	35.69
	Floor Pl.	0.5027	0.5016	0.5849	0.4703	S	0.0552	54.41
Federal Environmental Agency, Dessau	Facade	0.0697	0.1245	0.1108	0.8983	ET	0.0228	121.40
	Floor Pl.	0.1829	0.2208	0.2333	0.7877	S	0.0210	128.74
de Young Museum, San Francisco	Facade	0.1735	0.2607	0.2055	0.7868	ET	0.0475	65.00
	Floor Pl.	0.5005	0.6006	0.5954	0.4345	ET	0.0351	88.10
Des Moines Public Library, Iowa	Facade	0.1568	0.2625	0.1896	0.7970	ET	0.0595	44.18
	Floor Pl.	0.2228	0.3554	0.2537	0.7227	ET	0.0781	27.60
Kunsthaus, Austria	Facade	0.3450	0.5258	0.4393	0.5633	ET	0.0891	4.54
	Floor Pl.	0.5456	0.5000	0.5674	0.4623	S	0.0297	74.44
Mean	Facade	0.3444	0.4679	0.4413	0.5821	-	0.0932	28.98
	Floor Pl.	0.5572	0.4899	0.6078	0.4488		0.1001	64.78
	Total	0.4501	0.4789	0.5246	0.5155		0.0967	46.88

Figure 13. Circularity measured using the C10 method with reference circle MCC

Figure 14. Triangularity measured using the T6 method with the reference equilateral-triangle RET

Figure 15. Triangularity measured using the T6 method with the reference equilateral-triangle RET