Breaking the Plane: Exploring Real-Time Visualization of 3D Surfaces in Augmented Reality with Handwritten Input

Andr\'es Monroy-Hern\'andez; Beza Desta; Brian Lou; Jenny Sun; Kristoffer Selberg; Liam Franco Esparraguera; Parastoo Abtahi

REVIEW 3 major objections 4 minor 14 references

Breaking the Plane: Exploring Real-Time Visualization of 3D Surfaces in Augmented Reality with Handwritten Input

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read An AR headset that accepts handwritten equations and plots their 3D surfaces was rated more engaging than desktop and mobile graphing tools, while matching the desktop tool's ease of use.

desk verdict A genuinely exploratory AR prototype with an honest limitations section, but the Wizard-of-Oz parser means the headline claim of 'handwritten equation parsing' is not actually tested. read the letter →

arxiv 2507.05616 v1 pith:VIKA3L2Z submitted 2025-07-08 cs.HC

Liam Franco Esparraguera , Kristoffer Selberg , Brian Lou , Jenny Sun , Beza Desta , Andrés Monroy-Hernández , Parastoo Abtahi This is my paper · ORCID

classification cs.HC

keywords augmentedrealityhandwritteninputequationparsing3Dfunctionvisualizationhead-mounteddisplayWizard-of-Ozmathematicseducationuserstudy

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that combining handwritten equation input with real-time 3D surface rendering on an augmented-reality headset produces a more engaging and equally usable mathematics visualization tool than existing options. The authors built Breaking the Plane, a headset application that detects an equation written on a whiteboard, plots its 3D graph in the user's field of view, and lets the user grab, rotate, and rescale the graph. In a within-subjects study with 10 multivariable-calculus students, the system scored significantly higher on engagement than a plain whiteboard, GeoGebra AR, and GeoGebra Desktop; it matched GeoGebra Desktop on ease of use; and it was most often ranked as the most effective aid for the problem-solving questions. Users also reported a strong willingness to use the system again, with the handwriting input and direct graph manipulation cited as the reasons. The paper presents this as evidence that head-mounted AR with handwriting-style input can make mathematical visualization both more engaging and no harder to use than established desktop tools, while acknowledging that the parsing was simulated by a human operator.

What carries the argument

The load-bearing mechanism is the input loop: a user writes an equation on a whiteboard; an operator watching the headset's camera feed types it into the system over a WebSocket, with an "OCR Processing..." indicator masking the delay; a custom plotter evaluates the string expression and generates a procedural 3D mesh using the NCalc expression-evaluation library; and controller-based gestures let the user grab, rotate, scale, and pan the graph in the passthrough environment. The paper attributes the engagement and ease-of-use results primarily to this loop, because handwritten input accepts any notation a user already knows and frees the user from learning app-specific syntax.

What would settle it

Rerun the study with an actual handwriting recognizer running on the headset instead of the human operator; if engagement or ease-of-use ratings drop to the level of GeoGebra AR or GeoGebra Desktop, or if users report frustration with recognition latency and errors, the central claim would be refuted.

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Extended reading notes

Core claim

The paper introduces Breaking the Plane, an AR headset application that turns a handwritten two-variable equation into an interactive 3D surface plot rendered in the user's physical space. Its central finding is that this combination of handwritten input and head-mounted AR visualization was rated significantly more engaging than a plain whiteboard, GeoGebra AR, and GeoGebra Desktop, while being perceived as comparable in ease of use to GeoGebra Desktop. Participants ranked the system most often as the most effective tool for aiding problem-solving, and reported a high likelihood of future use. The authors interpret these results as evidence that HMD AR with handwriting-style input can support in-situ exploration of multivariable functions and remove the syntax barrier associated with existing graphing tools. The paper is careful to note that the parsing pipeline was simulated by a Wizard-of-Oz operator rather than an automatic recognizer.

Load-bearing premise

The study assumes that a human operator typing the user's handwritten equations into the system feels to the user like real automatic handwriting recognition, so that the measured engagement and ease-of-use ratings would carry over to a production OCR system.

Editorial extensions

If this is right

A head-mounted AR graphing tool can be as easy to use as a mature desktop application while being significantly more engaging, suggesting that AR need not trade usability for immersion.
Handwritten, syntax-free equation input removes the keystroke-learning barrier that prior work linked to low confidence with desktop graphing software.
Because participants ranked the system first or second for aiding problem-solving, a working automatic recognizer could make AR graphing a practical study aid for multivariable calculus.
The engagement advantage over a mobile AR app points to head-mounted, hands-free interaction, rather than AR itself, as the driver of the benefit.
The strong willingness-to-reuse scores suggest feasible adoption in educational settings if a production-ready handwriting recognizer is integrated.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

I infer that part of the engagement gain may be a novelty effect of the headset plus the operator's near-perfect transcription; a production recognizer with visible latency and errors could narrow the gap.
I infer that the handwriting-input mechanism could extend to other notation-heavy fields such as physics or engineering, where students sketch equations and diagrams that a recognizer could turn into manipulable 3D objects.
I infer that a longitudinal or between-subjects replication would test whether the engagement advantage persists after users become accustomed to the headset and the novelty fades.
I infer that the comparable ease-of-use result suggests input method, not display technology, is the main usability bottleneck for AR math tools; improving recognition may matter more than further polishing graph rendering.

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Signed reviews

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a small, honest exploratory study of an AR headset for 3D math surfaces with handwriting-style input, and the handwriting recognition is done by a human operator behind a loading spinner. The paper has a real prototype, clear writing, and a candid limitations section. But the abstract sells the Wizard-of-Oz condition as a working OCR pipeline, and the engagement and ease-of-use measurements were all taken under that simulated condition. So the real contribution is a design-space exploration, not a validated input method.

What is actually new: combining HMD AR with handwriting-like equation input for 3D surfaces. The cited Augmented Math work does OCR on mobile for 2D plots, and GeoGebra requires manual equation entry; neither offers this combination. The 3D function plotter and graph manipulation are implemented, and the comparison against a whiteboard and two GeoGebra variants is a reasonable baseline. It is believable that a headset with grab-and-move interaction is more engaging than a whiteboard. The authors also deserve credit for explicitly saying in Sec. 3.3 that they simulated the recognizer and in Sec. 7 that operator variability may have affected perceived ease of use.

Now the soft spots, in proportion. The load-bearing assumption is that the WoZ operator is a fair stand-in for real OCR. Participants never experienced a misread, never corrected an equation, and the latency was operator-dependent. That means the 4.6/5 rating for handwritten-equation input and the ease-of-use parity with GeoGebra Desktop are conditional on a perfect recognizer. A real OCR system will misread, require correction loops, and add variable delay, all of which could erode the measured advantages. The abstract says "real-time handwritten equation parsing" and "most effective in aiding problem-solving" without these caveats. The effectiveness result is also a self-reported ranking, not observed problem-solving performance, and the paired t-tests on 10 participants with no multiple-comparison correction are thin. The sample is homogeneous and the problems were not graded.

None of this is fatal for a workshop-format exploration. The paper is what it says in the introduction: a Wizard-of-Oz prototype. But the claims need to be reframed to match the evidence. For this extended abstract, the authors should at least adjust the title and abstract to disclose the simulation up front and release the questionnaire items. A serious referee could help them do that.

Who this is for: HCI people working on AR education and anyone interested in how Wizard-of-Oz prototyping interacts with user studies. I would not cite it in my own work without a real OCR follow-up, but I would send it to peer review rather than desk reject — the prototype is useful, the limitations are acknowledged, and the study can be improved with relatively modest changes.

Referee Report

3 major / 4 minor

Summary. The paper presents Breaking the Plane, an AR headset application (Meta Quest 3/Unity) intended to let users visualize 3D mathematical surfaces by writing equations by hand, with real-time graph manipulation and a custom 3D function plotter. The handwriting input is implemented as a Wizard-of-Oz pipeline in which a human operator views the headset camera feed and manually enters the equation, with a fake 'OCR Processing...' indicator masking the delay. A within-subjects user study (n=10, multivariate calculus students) compares four conditions: whiteboard, GeoGebra AR, GeoGebra Desktop, and Breaking the Plane, using Likert-scale ratings and ranking questions. The authors report significantly higher engagement for their system over all baselines, ease of use comparable to GeoGebra Desktop, the highest frequency of being ranked most effective at aiding problem-solving, and strong preference for future use.

Significance. If the findings held as stated, the contribution would be a useful demonstration that HMD-based AR with handwriting-style input can rival desktop tools in perceived usability while increasing engagement, addressing the content-authoring friction that prior AR math tools report. The paper has strengths: it compares against three external baselines, uses randomized system-query pairings, uses a within-subjects design, and transparently discloses the Wizard-of-Oz limitation, small sample, and learning effects. However, the empirical basis is thin and partly conditional on a simulated component; the headline claims overstate what was actually tested. As an exploratory extended-abstract study, the work is worth reporting, but the central claims need reframing to match the evidence.

major comments (3)

[§3.3] The core input capability, 'handwritten equation parsing,' is not implemented or tested: a human Wizard manually transcribes the equation over a WebSocket, and the user sees a simulated 'OCR Processing...' indicator. All input-related ratings, including the 4.6/5 contribution of handwritten input in §5.4, were collected under a condition with effectively perfect recognition and no error-correction loops. The paper acknowledges this in §7 ('introduced variability to our evaluation due to inconsistencies in operator performance'), but the Abstract and §8 present the system as having 'handwritten equation-parsing input' without this caveat. Since a real OCR parser would add latency, misrecognition, and correction friction—exactly the factors that could erode the measured engagement and ease-of-use advantages—the central claim as stated is not supported. Please either reframe all headline findings as applying to a Wizard-of-Oz-simulated handwriting interface, or supply evidence (e.g., operator transcription accuracy/latency logs and a sensitivity analysis) to justify generalizing to a real recognizer.
[§5] The inferential statistics rely on multiple paired t-tests on 5-point Likert items without correction for multiple comparisons. For the engagement comparisons, three pairwise tests are reported; under a simple Bonferroni correction (α = 0.05/3 ≈ 0.0167), the Breaking the Plane vs. GeoGebra AR comparison (p = 0.022) no longer meets the threshold, so the Abstract's 'significantly surpassed other tools' is not robust to standard multiple-comparison control. The ease-of-use section also reports several pairwise tests without familywise error control. Please report corrected p-values or false-discovery-rate adjustments, and include effect sizes or confidence intervals for the pairwise differences.
[§4.2.1] Problem-solving effectiveness is measured only by self-report, not by task performance. The study questions were explicitly 'not graded for correctness' (§4.2.1), and the reported evidence is a ranking (6 first, 4 second) plus retrospective Likert ratings. The claim that the system was 'the most effective in aiding problem-solving' should therefore be worded as perceived effectiveness, or supported by objective measures such as the number of correct answers or time-to-solution across conditions.

minor comments (4)

[§5.1] The sentence 'The t-test results provided in Fig. 3 demonstrate significant variances in ease of use' should say 'significant differences' rather than 'variances,' since the analysis concerns means, not variances.
[§5.2] The t-statistics for engagement are all negative; the sign convention should be stated explicitly (e.g., negative values indicate higher mean engagement for 'Our System') so readers can interpret the direction of the effects.
[Fig. 4] Figure 4, comparing 'our system with GeoGebra Desktop and GeoGebra AR in aiding problem-solving,' is mentioned only briefly in the text; the figure's axes, scales, and what the plotted values represent should be described in the caption or in §5.3.
[§3] The phrase 'the Meta Quest 3 is an AR-capable headset featuring color passthrough' is potentially confusing because the device is also marketed as a VR headset; consider using 'mixed reality' or 'passthrough-based AR' for precision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper reports an empirical within-subjects comparison against external baselines, and no result is derived from, or fitted to, its own inputs.

full rationale

Breaking the Plane is an empirical user study rather than a derivation. The central results—engagement, ease of use, problem-solving effectiveness, and future-use preference—are self-reported Likert-scale and ranking measures collected in a within-subjects study comparing the system to a whiteboard, GeoGebra AR, and GeoGebra Desktop. These outcome measures are not defined in terms of the paper's own assumptions or fitted parameters, and no quantity is fitted to a subset of data and then renamed as a prediction. The related-work citations are not load-bearing for the empirical findings; the comparisons are against external, independently developed tools. The most substantive concern, raised explicitly in Sec. 3.3 and Sec. 7, is that equation parsing was simulated by a Wizard-of-Oz operator rather than an actual OCR system, so the measured experience is conditional on a human mediator and the abstract's wording overstates what was tested. That is a threat to external validity and a limitation, but it is not circularity: the user-perception data do not reduce by construction to the study's design choices, and the paper does not claim to derive handwriting-recognition accuracy from the participants' ratings. Accordingly, no circular step can be quoted, and the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim is an empirical user-study result. It relies on assumptions about the Wizard-of-Oz setup as a stand-in for real OCR, on treated Likert data as interval measurements, on the adequacy of randomization controls, and on the sufficiency of n=10. There are no fitted numerical parameters and no newly posited physical or technical entities.

assumptions (4)

ad hoc to paper A Wizard-of-Oz operator can substitute for a real handwriting OCR system in evaluating perceived ease of use and engagement.
The study evaluates the system with a human operator entering equations (Sec. 3.3); if operator performance differs from an actual recognizer, the engagement and ease-of-use results do not transfer to a real pipeline.
domain assumption Self-reported Likert ratings measure engagement, ease of use, and effectiveness in aiding problem-solving.
The central statistics are paired t-tests over 5-point Likert responses (Sec. 5.1-5.2), which assumes these subjective ratings are interval-like and capture the constructs claimed.
domain assumption Random assignment of system-surface pairings and within-subjects ordering adequately controls learning effects.
The study uses random sequences to counter order effects (Sec. 4.2), but participants reported remembering visualizations from earlier rounds (Sec. 7), so the control may be incomplete.
domain assumption 10 participants provide enough statistical power for the paired comparisons.
n=10 with paired t-tests and no multiple-comparison correction (Sec. 5) makes the significant differences fragile; the paper acknowledges the sample's limits (Sec. 7).

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Cite this review

Pith. "Pith review of Breaking the Plane: Exploring Real-Time Visualization of 3D Surfaces in Augmented Reality with Handwritten Input." pith.science (2026). https://pith.science/paper/VIKA3L2Z

@misc{pith2026250705616,
  author       = {Pith},
  title        = {Pith review of: Breaking the Plane: Exploring Real-Time Visualization of 3D Surfaces in Augmented Reality with Handwritten Input},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VIKA3L2Z}},
  note         = {Machine review of arXiv:2507.05616}
}

read the original abstract

We introduce Breaking the Plane, an augmented reality (AR) application built for AR headsets that enables users to visualize 3D mathematical functions using handwritten input. Researchers have demonstrated overlaying 3D visualizations of mathematical concepts through AR enhances learning motivation and comprehension, and equation parsing makes the authoring of teaching materials more time-efficient for instructors. Previous works have developed AR systems that separately employ equation parsing and 3D mathematical visualizations, but work has yet to be done to combine those features by enabling real-time interactions and dynamic visualizations that help users learn in situ. We explore this by developing an AR system featuring handwritten equation parsing, graph manipulation, and a 3D function plotter. We found that our system significantly surpassed other systems in engagement, achieved comparable ease of use to a popular visualization tool, was considered the most effective in aiding problem-solving, and was highly preferred by participants for future use.

Figures

Figures reproduced from arXiv: 2507.05616 by the authors.

**Figure 2.** Our technical systems in the order: GeoGebra AR, [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Graphs comparing user response on the perceived [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Direct comparison of our system with GeoGebra [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

Mehmet Bulut and Rita Borromeo Ferri. 2023. A Systematic Literature Review on Augmented Reality in Mathematics Education. European Journal of Science and Mathematics Education 11, 3 (2023), 556–572

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[3]

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Mehmet Bulut and Rita Borromeo Ferri. 2023. A Systematic Literature Review on Augmented Reality in Mathematics Education. European Journal of Science and Mathematics Education 11, 3 (2023), 556–572

work page 2023

[2] [2]

Yeliz Celen. 2020. Student Opinions on the Use of Geogebra Software in Mathe- matics Teaching. Turkish Online Journal of Educational Technology-TOJET 19, 4 (2020), 84–88

work page 2020

[3] [3]

Hsin-Yi Chang, Theerapong Binali, Jyh-Chong Liang, Guo-Li Chiou, Kun-Hung Cheng, Silvia Wen-Yu Lee, and Chin-Chung Tsai. 2022. Ten years of augmented re- ality in education: A meta-analysis of (quasi-) experimental studies to investigate the impact. Computers & Education 191 (2022), 104641

work page 2022

[4] [4]

Neil Chulpongsatorn, Mille Skovhus Lunding, Nishan Soni, and Ryo Suzuki. 2023. Augmented Math: Authoring AR-Based Explorable Explanations by Augmenting Static Math Textbooks. arXiv preprint arXiv:2307.16112 (2023)

work page Pith review arXiv 2023

[5] [5]

Manoela Milena Oliveira da Silva, Rafael Alves Roberto, Iulian Radu, Patri- cia Smith Cavalcante, and Veronica Teichrieb. 2019. Why Don’t We See More of Augmented Reality in Schools?. In 2019 IEEE International Symposium on Mixed and Augmented Reality Adjunct (ISMAR-Adjunct). IEEE, 138–143

work page 2019

[6] [6]

Shubham Gargrish, Deepti P Kaur, Archana Mantri, Gurjinder Singh, and Bhanu Sharma. 2021. Measuring effectiveness of augmented reality-based geometry learning assistant on memory retention abilities of the students in 3D geometry. Computer Applications in Engineering Education 29, 6 (2021), 1811–1824

work page 2021

[7] [7]

Angel Gutierrez, John Pegg, and Christine Lawrie. 2004. Characterization of Students’ Reasoning and Proof Abilities in 3-Dimensional Geometry.International Group for the Psychology of Mathematics Education (2004)

work page 2004

[8] [8]

Hartini Ismail, Abdul Halim Abdullah, N Syuhada, and NH Noh. 2020. Inves- tigating student’s learning difficulties in Shape and Space topic: A case study. International Journal of Psychosocial Rehabilitation 24, 5 (2020), 5315–5321

work page 2020