This guest blog explores the relationship between incisor movement and space creation or utilization. It highlights why understanding the link between incisor movement and space requirements is crucial for everyday clinical decision-making. While the principles behind incisor positioning and its impact on space requirements have guided clinicians for decades, they remain scattered across research and clinical training. This is the third blog post in a series called the Mathematics of Orthodontics, which aims to bring those principles together into a consolidated framework.
This series is written by Daniel Richmond, an orthodontist from Ontario, Canada, who I came across on social media. Daniel has put together an innovative online resource for orthodontic space analysis, the Ortho Calculator, which I found interesting and very useful in terms of ensuring a structured and comprehensive orthodontic assessment for space planning (I have no conflict of interest).
With the ideals for planned incisor position (PIP) established in Blog Post 2, it becomes essential to evaluate how incisor movement influences the space analysis. A useful starting point is to consider the geometric relationship between incisor inclination and anteroposterior (AP) movement. According to McLaughlin, Bennett, and Trevisi (MBT), every 2.5° of proclination corresponds to a 1 mm forward movement of a lower incisor’s incisal edge (McLaughlin et al., 2001). But where does this ratio come from?
Assuming the average upper or lower incisor is approximately 22 mm in length, the movement can be modelled as controlled tipping using an isosceles triangle: the equal sides represent the tooth’s initial and final positions, and the base represents the AP displacement of the incisal edge. This geometric framework links angular changes in inclination to linear displacement of the incisal edge, as illustrated in Figure 1.
Figure 1. (A) Clinical example illustrating a 1 mm forward displacement of the incisal edge following proclination. (B) Equivalent isosceles triangle model (tooth length 22 mm), demonstrating that a 2.6° change in inclination corresponds to 1 mm of AP displacement of the incisal edge. Diagrams not to scale; provided for illustrative purposes.
This geometric model yields a ratio of 2.6° of inclination change per 1 mm of AP displacement, very similar to the 2.5° per 1 mm ratio proposed by MBT. Real-world data supports this relationship. In their study of crowding resolution, Yitschaky et al. (2016) found that for every millimeter of crowding resolved, the lower incisors proclined by an average of 0.5° and protruded by 0.2 mm. This equates to roughly 2.5° of proclination per 1 mm of protrusion. From these findings, a general rule emerges:
Building on this geometric model, the next consideration is the impact of incisor movement on space analysis. Foundational cephalometric literature has long suggested that changes in the AP position of the incisors significantly affect arch perimeter. Pioneers in orthodontics, such as Steiner and Ricketts, proposed that for every 1 mm of AP incisor movement, arch perimeter changes by approximately 2 mm (Ricketts, 1982; Steiner, 1960). The Royal London Space Planning system offers similar guidance, recommending that clinicians account for a 2 mm change in arch perimeter for every 1 mm of AP incisal edge movement (Kirschen et al., 2000). Although these estimates were not derived from controlled experimental studies, they have historically shaped clinical intuition and treatment planning.
More recent research has sought to validate these assumptions through geometric modelling. Using elliptical representations of dental arches, Aghera et al. (2016) found that 1 mm of incisal edge advancement generated an average arch perimeter gain of 1.67 mm in the maxilla and 1.65 mm in the mandible. Similarly, Chung and Wolfgramm (2015) estimated that each 1 mm of maxillary incisor protrusion produced, on average, a 1.66 mm increase in arch perimeter. Mutinelli et al. (2000) reported that 1 mm of AP incisal edge movement could lead to 1.2–2.1mm of arch perimeter change, depending on arch form, with an average change of 1.52 mm. Collectively, both classic estimates and modern models support the following guideline:
Figure 2. Relationship between AP incisal edge movement and arch perimeter. For every 1 mm of anterior incisal edge movement (red), there is an associated 1.5–2.0 mm increase in arch perimeter (blue); the reverse relationship applies during incisor retraction. Diagram is not to scale; provided for illustrative purposes.
While AP incisal edge displacement reflects the combined effects of tipping, torque, and translation, several studies have attempted to isolate inclination alone and relate it directly to arch perimeter changes. The Royal London Space Planning system offers a conservative estimate of 0.2 mm of space created per degree of proclination (Kirschen et al., 2000). Using geometric models of different arch forms, Mutinelli et al., (2000) reported a broader range of 0.34–0.74 mm of space gain per degree of proclination, with controlled tipping producing more than uncontrolled tipping. At the higher end, Tweed (1954) and MBT (2001) suggest values of 0.8 mm of space gain per degree of proclination.
By comparison, the isosceles triangle model described earlier predicts that 1° of proclination produces 0.4 mm of AP incisal edge advancement. Applying the AP arch perimeter change formula (AP movement × 1.5–2.0), this displacement translates into 0.6–0.8 mm of space gain. This estimate is consistent with the relationship proposed by Tweed and McLaughlin and falls within the range reported by Mutinelli. Synthesizing this information, the relationship can be summarized as follows:
The most intuitive and reliable parameter to use in space analysis calculations is AP incisal edge displacement. This single measure captures both linear and angular components of incisor movement in a unified way, integrating tipping, torque, and translation without the need to separate them. Because the exact balance of these movements is rarely predictable, even with carefully planned mechanics (Elnagar et al., 2021; Li et al., 2015; Ruan et al., 2018), the most practical approach is to plan around the final AP position of the incisal edge itself. This method allows cephalometric and clinical objectives to be incorporated seamlessly while remaining adaptable to variations in skeletal base relationships, as discussed in Blog Post 2. Ultimately, it is the AP displacement, not the specific type of movement, that dictates arch perimeter change. For this reason, I propose adopting the already established relationship as the most logical basis for treatment planning. This relationship has been frequently described, without a clear formula, I’ll refer to it here as the Edge-to-Arch formula (Richmond, 2025), a label used simply for clarity in clinical discussions.
The Edge-to-Arch formula predicts that each millimeter of AP incisal edge advancement or retraction yields 1.5–2.0 mm of arch perimeter change. But which end of this range best reflects clinical practice? Consider the upper arch of a Class II extraction case with 4 mm of crowding and 5 mm of retraction needed to reach PIP (Figure 3). Running the numbers with the lower (1.5) versus higher (2.0) multiplier of the Edge-to-Arch formula results in a 2.5 mm difference in total space required (Table 1).
Figure 3. Class II extraction case. (A) Maxillary arch with 4 mm of crowding to resolve. (B) Cephalometric tracing showing the maxillary incisors positioned 5 mm ahead of ideal (relative to the NA line), requiring retraction to achieve PIP.
Requirement | 1.5 Multiplier | 2.0 Multiplier |
To resolve crowding | 4 mm | 4 mm |
To facilitate retraction | 7.5 mm | 10 mm |
Total space required | 11.5 mm | 14 mm |
Difference | 2.5 mm | |
Table 1. Space requirements to resolve crowding and facilitate incisor retraction. Space requirements for retraction were calculated using the 1.5× and 2.0× multipliers of the Edge-to-Arch Formula.
In a specialty of millimeters, a 2.5 mm difference is clinically meaningful. Even with maximum anchorage mechanics, 1–2 mm of anchorage loss per side (2–4 mm total) is common (Rajesh et al., 2014; Su et al., 2014; Thiruvenkatachari et al., 2006; Yassir et al., 2019). This expected anchorage loss (~3 mm) closely matches the difference between the low and high estimates (Table 1). By using the larger multiplier of 2.0, the calculation provides a built-in safety margin that accounts for anchorage loss and mechanical inefficiencies commonly observed in clinical practice. This minimizes the risk of underestimating space requirements and ensures that extractions or other space-gaining strategies are planned appropriately.
Final recommendation for treatment planning:
Planned incisor movement can dramatically reshape the outcome of the baseline space analysis outlined in Blog Post 1. A case that initially appears severely crowded may be treated non-extraction if 4 mm of planned incisor protrusion provides 8 mm of additional arch perimeter. Conversely, a case that seems only mildly crowded may become severely space deficient if 5 mm of retraction demands 10 mm of space. This reinforces a key principle: space analysis is not static. It is a multi-step process that must account for both the presenting malocclusion as well as the planned final tooth positions. Starting with the end in mind is essential for math-based treatment planning.
The next post will address arch coordination in the transverse plane and its role in space creation. Subsequent posts will cover additional strategies to ensure the numbers behind your plan align with your treatment objectives.
Contents: Daniel Richmond 
Visit the Ortho Calculator at www.orthocalculator.org/
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