Objective Based on the visual cone model, the field of view is a conical range formed by the convergence of countless lines of sight. Therefore, there is always a portion of the proximal spatial depth near the viewpoint that lies outside the field of view and cannot be visually captured, resulting in “proximal engulfment”. Proximal engulfment, an unavoidable visual phenomenon, relates to how landscape elements near the observer appear. Its impact on spatial arrangement design in Western formal gardens remains underexplored.

Methods Building on the well-established understanding of perspective through the visual cone among Western craftsmen by the end of the 15^th century, this research employs the visual cone to establish the core geometric mechanism of proximal engulfment phenomenon under horizontal viewing conditions, proposing L_PE (the horizontal ground depth length in front of the viewpoint that is proximally engulfed) as a measurement parameter. Through combined spatial structure analysis and scene presentation simulation within case studies, design responses to proximal engulfment in spatial construction can be extracted. The geometric relationship between proximal engulfment and the visual cone reveals L_PE’s dependence on h (viewpoint height above ground) and θ (sight-line depression angle), with L_PE=h·cot(θ). Considering that the vertical natural field of vision of the human eye is around 35 degrees, the value of θ can be 17.5 degrees, which means the value of L_PE is approximately 3.17h. Thus, with the known viewpoint height above the ground (h), it is possible to calculate the horizontal proximal engulfment depth L_PE corresponding to the viewpoint. By comparing d_N (the distance from the viewpoint to the near endpoint of the spatial unit) and d_F (distance from the viewpoint to the far endpoint of the spatial unit) with this parameter, the visibility of the corresponding spatial unit in the horizontal view from the viewpoint can be determined: If d_N>L_PE, the spatial unit is fully visible; if d_N<L_PE<d_F, the spatial unit is partly visible; if d_F<L_PE, the spatial unit is totally invisible. The Cortile del Belvedere, dating to the early 16^th century, is widely recognized in the field of garden history as an important prototype of formal gardens. It is probable that the perceptual impact and design strategies observed in this case reflect similarities to those found in Western formal garden design practices during the 16^th and 17^th centuries, based on the designer’s understanding of proximal engulfment. Thus, by using L_PE as a parameter in visual coverage analysis and scene presentation simulation in this case, and the comparative analysis combining the examples of Villa Farnese and Tuileries Garden, manifestations of proximal engulfment in Western formal garden spatial perception and arrangement can be clearly identified.

Results L_PE can quantitatively describe the extent of the proximal engulfment that occurs on the ground of the observed spatial unit under the condition of horizontal viewing from the corresponding viewpoint. Based on this parameter, case analysis have clarified that when there is a height difference between the viewpoint and the ground of the frontal spatial sequence, it is difficult to avoid three perceptual effects caused by proximal engulfment: compression of scene depth perception, visual truncation of unit landscape, and visual separation of near and far spaces. The degree of these impacts is proportional to the height of the viewpoint above the ground. Based on the geometric mechanisms of proximal engulfment, garden designers can regulate the degree of perceptual impacts caused by proximal engulfment in spatial construction by coordinating the planar position, height difference, and depth of the viewing space units in relation to the viewpoint. As design strategies, concealing functional spaces, highlighting distinctive scenic elements, and enhancing the viewing hierarchy correspond respectively to these three impacts. Additionally, the bidirectional analytical framework established based on the L_PE enables forward prediction of “spatial data−visual presentation” and reverse derivation of “historical imagery−viewpoint parameters”, offering a novel approach for digital research on historic gardens.

Conclusion This study clarifies the geometric mechanism of proximal engulfment, analyzes its impact on the spatial perception of Western formal gardens through case studies, and elaborates on the design strategies in spatial construction in response to these impacts. The study proposes a quantitative method for evaluating the horizontal visibility of spatial units using the horizontal proximal engulfment depth L_PE. By comparing the height of the viewpoint above the ground with the viewing distances of the near and far endpoints of the spatial unit, this method allows for precise determination of the visibility and manner of appearance of the spatial unit from the viewpoint. The analytical methods and the corresponding relationships between impacts and strategies proposed in this study can be used to analyze the spatial construction characteristics of other Western formal gardens. By comparing the similarities and differences in techniques across different cases, this approach not only deepens the understanding of the design principles of Western formal gardens but also provides historical references for the rational construction of contemporary landscape spaces based on visual principles.