Master Telescope Collimation and Star Testing

Table of Contents

• Introduction
• Why Collimation Matters
• Essential Tools and the Star Test
• Collimating Newtonian Reflectors
• Collimating Schmidt–Cassegrains (SCT/EdgeHD)
• Collimating RC and Classical Cassegrain Telescopes
• Star Testing and Diagnosing Aberrations
• Mechanical Alignment and Upgrades
• Collimation for Visual vs Imaging
• Common Mistakes and How to Avoid Them
• Transport, Thermal Management, and Maintenance
• Field Checklist: Collimation in 5–10 Minutes
• FAQs
• Advanced FAQs
• Conclusion and Next Steps

Introduction

Collimation is the geometric alignment of a telescope’s optical elements so that light travels symmetrically to the focal plane. Even a modest misalignment can soften detail, reduce contrast, increase aberrations off-axis, and waste the resolving power you paid for. While the word can sound intimidating, collimation is a learnable skill. With the right tools and a repeatable workflow, you can bring a Newtonian, Schmidt–Cassegrain (SCT/EdgeHD), or Ritchey–Chrétien (RC)/Classical Cassegrain (CC) to peak performance in minutes.

This guide covers practical, step-by-step procedures, the star test for verification, and diagnostics for common optical issues. If you’re new to the process, focus on the essentials in Essential Tools and the Star Test and the design-specific section for your telescope (for example, Newtonian Reflectors or SCT/EdgeHD). If you image, don’t miss Collimation for Visual vs Imaging, where we discuss tilt, sensor orthogonality, and critical tolerances for fast systems.

Why Collimation Matters

Optical systems deliver their best performance only when their components are coaxial and properly spaced. Misalignment results in asymmetry of the point spread function (PSF) and increases off-axis aberrations such as coma and astigmatism. The practical consequences include:

• Soft planetary detail: Low-contrast features like belt rifts on Jupiter and albedo markings on Mars are the first casualties of misalignment.
• Bloated stars: The Airy disk and diffraction rings become distorted or smeared, enlarging stars and reducing separation in double star observing.
• Uneven field illumination: Secondary obstruction and focuser misalignment can cause vignetting or uneven flats in imaging.
• Field-dependent issues: Coma correctors and flatteners assume proper alignment; miscollimation makes their corrections imperfect, especially noticeable in fast systems.

Collimation tolerances depend on focal ratio. Faster systems (e.g., f/3–f/5 Newtonians or f/2 HyperStar) are less forgiving and require more precise alignment than slower ones (e.g., f/10 SCTs). As a rule of thumb, acceptable primary mirror axial error in a Newtonian is on the order of a few percent of the primary diameter, shrinking quickly at fast f/ratios. In practice, that means a well calibrated tool and careful procedure become essential as speed increases.

Finally, remember that seeing, thermal equilibrium, and tracking can mimic or mask collimation errors. The star test (see Star Testing and Diagnosing Aberrations) is the gold standard for verification, but it must be performed under suitable conditions.

Essential Tools and the Star Test

You don’t need a suitcase of gear to collimate effectively, but a few tools make the process faster and more reliable.

Core toolkit

• Collimation cap: A simple peep-hole cap that helps center the primary mirror’s center spot in Newtonians. Cheap and effective.
• Cheshire/sight tube: Combines a peephole and a bright reflective surface to align the primary mirror and set secondary tilt and placement. Very effective for Newtonians.
• Laser collimator: Useful for aligning the Newtonian secondary to the focuser axis. Must be collimated itself; a misaligned laser can mislead.
• Barlowed laser attachment: Projects a silhouette of the primary’s center mark back to the screen, making primary alignment insensitive to laser registration errors.
• Phillips/Allen drivers: For adjusting mirror cells, secondary holders, or SCT secondary screws. Use the correct tool to avoid damage.
• Artificial star: A small, bright point source for indoor/yard testing at long distance (ideally ≥ 20× focal length). Useful when seeing is poor.

Helpful accessories

• Center-spotted primary (Newtonian): A small paper/foil donut to mark the optical center. Modern mirrors usually come marked.
• Secondary center marker/template: For Newtonians when re-centering is required after hardware changes.
• Focuser tilt/rotator with shims: For imaging rigs where the camera sensor must be square to the optical axis.
• Dew shield and heaters (SCT/Correctors): While this guide isn’t about dew control, preventing dew maintains visibility of diffraction patterns during star tests.

The star test: your final arbiter

The star test is a simple, sensitive method to verify collimation and assess optical quality:

1. Pick a suitable star: Choose a bright star near zenith to minimize atmospheric dispersion and seeing. Polaris is convenient because it barely drifts.
2. Thermal equilibrium: Allow the telescope to reach ambient temperature; thermal plumes can resemble aberrations.
3. High magnification: Use 25–50× per inch of aperture (1–2× per mm). For a 200 mm scope, 200–400× is typical when seeing allows.
4. Intra/extra focal patterns: Slightly defocus inside and outside of focus. Diffraction rings should be concentric and similar in appearance.
5. On-axis first: Collimate using an on-axis star; off-axis stars will show field aberrations from the design and are not suitable for alignment.

For Newtonians, finalize using a Barlowed laser or Cheshire and validate on a star. For SCT/EdgeHD and RC/CC, star collimation is the primary method; daytime rough alignment can save time, but final tweaks should be done under the stars.

We’ll refer back to the star test often, so if you need a refresher later, jump to Star Testing and Diagnosing Aberrations. Also, each design’s workflow references tools listed above—see Newtonian Reflectors, SCT/EdgeHD, and RC/CC for specifics.

Collimating Newtonian Reflectors

Newtonian collimation involves three nested tasks: (1) placing the secondary under the focuser (centering and offset), (2) aiming the secondary so the focuser axis hits the primary’s center, and (3) aiming the primary so its axis returns to the focuser/camera.

Know your geometry: offset matters

In fast Newtonians (e.g., f/4–f/5), the secondary mirror should be offset away from the focuser and toward the primary to evenly illuminate the field. The approximate offset is:

Secondary offset ≈ minor axis / (4 × f-ratio)

Many modern secondary holders build in this offset. Don’t chase perfect concentricity of every reflection; instead, aim for the correct geometry. A sight tube helps visualize secondary placement. For imaging, proper offset helps flat-fielding and edge-star uniformity.

Step-by-step: daylight alignment

1. Center and round the secondary (placement)
   • Insert a sight tube into the focuser. Rack the focuser to place the sight tube’s inner edge slightly larger than the secondary mirror outline.
   • Adjust secondary position (up/down the tube) and rotation until the secondary appears round and centered under the focuser. Expect a slight offset in fast scopes.
2. Aim the secondary (focuser axis)
   • Use a well-collimated laser collimator or the crosshairs of a sight tube.
   • Adjust the secondary tilt to place the laser dot on the primary center spot (or center the primary reflection in the crosshairs).
   • Re-check rotation; tilt and rotation interact. Iterate until the laser consistently hits the center spot.
3. Aim the primary (primary axis)
   • Use a Cheshire or a Barlowed laser. With a Barlowed laser, the returning shadow of the center mark should be centered on the laser’s face target.
   • Adjust the primary mirror’s collimation knobs a little at a time. Lock bolts (if present) should be snug but not distorting the mirror cell.
4. Verify with a star
   • At night, perform a star test. Fine-tune the primary if the donut shadow is off-center at slight defocus.

Special case: fast Newtonians (f/3–f/4)

• Tool quality is critical: Consider a quality autocollimator and well-calibrated laser. Small errors are magnified at fast f/ratios.
• Coma corrector alignment: Insert your corrector when checking backfocus and tilt for imaging. The corrector assumes axes are well aligned.
• Focuser squareness: Verify the focuser drawtube is orthogonal to the tube/OTA. If not, your secondary placement will never look right.

Troubleshooting Newtonians

• Coma on-axis: Indicates primary axis misalignment. Re-check primary with a Cheshire or Barlowed laser.
• Uneven illumination: Secondary too far toward/away from the focuser, or insufficient offset. Re-check secondary placement.
• Triangular stars: Often due to pinched optics—mirror clips too tight. Loosen so a business card can slide between clip and mirror.
• Astigmatism that rotates with focuser: Focuser drawtube tilt or eyepiece/camera/focuser compression misalignment. Use a compression ring or threaded connection; check focuser tilt.

Once you’re comfortable with this process, you can perform the daytime steps in minutes. For final precision, always verify with a star as described in Star Testing.

Collimating Schmidt–Cassegrains (SCT/EdgeHD)

SCTs and EdgeHDs are collimated by adjusting the secondary mirror tilt so that the optical axis is aligned to the baffle and primary. Mechanical spacing and primary tilt are factory-set; your only routine adjustment is the secondary screws.

Preparation

• Let the scope cool: Thermal plumes can ruin the diffraction pattern.
• Dew control: Keep the corrector plate clear; you must see the pattern. A dew shield helps and is advisable in humid conditions.
• Choose a star: Bright, near-zenith. Polaris is convenient for manual mounts.
• Magnification: Use a high-power eyepiece (200× or more when seeing allows). For imaging, you can use a camera with short exposures and ROI.

Rough collimation (if way off)

In daylight, aim at a distant glint or the reflection of the Sun on a distant insulator (never at the Sun). Rack in/out of focus to see the donut shadow. Adjust one secondary screw at a time to center the shadow in the bright disk. This only gets you close; finish under the stars.

Fine collimation on a star

1. Defocused donut: Slightly defocus to show a concentric bright ring pattern with a central dark shadow (secondary). If the shadow is off-center, pick the screw that moves the star toward the center of the field and turn it a small amount.
2. Iterate in small steps: Make tiny adjustments (eighth turns). Recenter the star after every adjustment.
3. Near-focus refinement: As you approach focus, the first diffraction ring should be symmetrical around the Airy disk. If a brightening appears on one side, adjust the screw that moves the star toward the bright side.
4. Final check at focus: On a steady night, the Airy disk should be round with symmetric rings. Slight seeing boil is normal.

EdgeHD and collimation

EdgeHD systems include internal lenses to flatten the field. The collimation method is the same as standard SCTs; however, because the imaging circle is corrected, precise on-axis collimation is particularly important for edge performance. For astrophotography, confirm symmetry of star shapes at all corners after collimation; if corners differ, check tilt and backfocus, not just collimation.

Notes on mechanics

• Mirror flop: Focus shift can mimic tilt. Some SCTs offer mirror locks; use them after focusing for imaging.
• Bob’s Knobs vs factory screws: Tool-less knobs make adjustments easier but can be more sensitive to unintended bumps. Use what you are comfortable with and adjust carefully.
• Secondary rotation: Do not loosen screws enough to allow the secondary to rotate unless instructed by the manufacturer; rotational clocking can matter in some systems.

If you are switching from visual to HyperStar (f/2 prime focus), be aware that tolerances become extremely tight. Collimation at f/2 may need to be verified with the camera attached, and even small sag can degrade stars. See Collimation for Visual vs Imaging for tilt considerations.

Collimating RC and Classical Cassegrain Telescopes

Ritchey–Chrétien (RC) and Classical Cassegrain (CC) systems are popular for imaging because they suppress some aberrations (e.g., RC eliminates on-axis coma), but they demand careful alignment. Many RCs feature a center-spotted primary and adjustable secondary; some include a focuser tilt plate.

Key principles

• Co-axial mirrors: Primary and secondary axes must coincide.
• Focuser orthogonality: The focuser (and thus camera sensor) must be square to the optical axis.
• Spacing matters: Primary–secondary distance (set by the truss/tube) affects spherical correction; do not change unless instructed by the manufacturer.

Workflow: daytime setup

1. Center the secondary: Use a sight tube or laser to ensure the secondary appears centered under the focuser. Adjust secondary position and rotation minimally; strive for symmetry.
2. Align the focuser to the primary (if tilt plate present): Using a laser that returns on itself, adjust the focuser tilt so the beam hits the primary center and returns to the collimator aperture.
3. Align the secondary to the primary: Now adjust the secondary tilt so the returning beam remains concentric while centered on the primary spot.
4. Primary adjustment: Some workflows recommend adjusting primary tilt last using a Cheshire to center the reflected images. Follow your manufacturer’s sequence; RCs can be iterative.

Verification: star collimation

• On-axis star: Use high magnification or a camera with short exposures. Defocus slightly to assess concentricity; then test near focus.
• Corner stars: For imaging, examine all four corners. If two opposite corners are elongated in opposite directions, suspect sensor tilt or focuser tilt, not mirror collimation.
• Repeat small adjustments: RC systems are sensitive; tiny tweaks can have large effects. Always recenter the star after each change.

Common RC/CC issues

• Unequal corner shapes: Sensor/focuser tilt or spacing/backfocus errors with the field flattener. Check mechanical orthogonality and exact backfocus spacing.
• Astigmatism on-axis: Could be residual miscollimation or mechanical stress in the mirrors. Confirm at multiple rotations; if it rotates with the camera, it’s likely tilt; if it stays fixed, revisit collimation.

Because RC/CC procedures vary by model, consult your manual for specifics on which mirror to adjust first and how to interpret reflections. The general approach above, combined with the star test, covers most cases.

Star Testing and Diagnosing Aberrations

The star test is a sensitive probe of alignment and optical quality. While comprehensive texts delve into wavefront analysis, you can glean much from the symmetry and sharpness of the diffraction pattern.

Setup for a reliable test

• Thermal equilibrium: Wait until tube currents subside. Fans (in some Newtonians) can speed this up.
• Steady seeing: Aim near zenith on nights of good seeing. If the pattern boils constantly, defer fine evaluation.
• On-axis star: Always test on-axis.
• Magnification: Use high power; you should see at least a few diffraction rings when slightly defocused.

Interpreting patterns

• Perfect collimation: Defocused intra- and extra-focal rings are concentric and similar; at focus, the Airy disk is round with a faint, even first ring.
• Decentering/tilt: The central shadow (secondary) is off-center in the donut; rings are brighter on one side. Remedy by adjusting collimation as per your design (SCT, Newtonian, RC/CC).
• Spherical aberration: Intra- and extra-focal patterns differ (e.g., inner ring brightness swaps sides). Some difference is normal; large differences suggest figure error or incorrect spacing (relevant to RC/CC).
• Astigmatism: At slight defocus, the pattern looks oval. When you focus through, the major axis of the oval rotates by ~90°. Could be optical stress or tilt; verify by rotating the eyepiece/camera to see if it rotates.
• Coma: Off-axis comatic stars are comet-shaped. On-axis coma indicates miscollimation (primary axis in Newtonians, secondary tilt in SCT/RC).
• Turned edge: The outer ring appears too bright and persistent; in focus, stars may show a bright rim. Hard to fix without masking; mild cases may be acceptable.
• Zonal errors: Uneven brightness in mid-rings intra vs extra; mild zones are common and often negligible visually.
• Thermal plumes: Wavering, moving asymmetries that change with time, often rising from the bottom of the field. Wait longer or manage cooling.

Artificial stars

An artificial star can be used for rough assessments and collimation when sky conditions are poor. Place it far away—ideally ≥ 20× the telescope’s focal length—to approximate infinity focus. Ensure the star is small and bright (e.g., a reflective bead LED source). Beware of ground-layer seeing over long distances near the surface; raising the target helps.

Using a camera for star testing

Short-exposure video (lucky imaging style) lets you average out seeing to view the Airy disk and rings. For SCT/RC imaging rigs, this can be the most practical way to collimate precisely. Always recenter the star after each adjustment; recentration errors can masquerade as miscollimation.

Mechanical Alignment and Upgrades

Optical alignment depends on the mechanics that hold mirrors and focusers. Good mechanics make collimation easy and stable; poor mechanics make it frustrating.

Focuser alignment and registration

• Orthogonality: The focuser should be square to the tube/optical axis. Shims or a tilt plate can correct small errors.
• Registration repeatability: Compression rings and threaded adapters improve repeatability vs set screws, which can skew the eyepiece/camera.
• Backfocus integrity: For imaging, verify spacing with calipers or spacers; sloppy spacing can exaggerate tilt/field errors.

Mirror cells and stress

• Primary mirror clips: Should not pinch. Leave a tiny gap (e.g., a business card) between clip and glass.
• Secondary holders: Check that the spider is tensioned evenly and doesn’t twist the vane assembly.
• Locking screws: On some Newtonians and SCTs, locks can help hold collimation but may induce stress if over-tightened.

Upgrades that help

• Dual-speed focuser: Fine focus control improves star test and collimation precision.
• Autocollimator (Newtonian): Amplifies small residual errors; excellent for fast scopes.
• Secondary knobs (SCT): Easier star collimation, but adjust carefully to avoid over-loosening.
• Tilt plate (imaging): Allows sensor orthogonality adjustments independent of mirror collimation.

Transport and stability

• Truss vs tube: Truss designs may require more frequent touch-ups; tubes often hold collimation better during transport.
• Temperature swings: Large gradients can shift alignment temporarily; recheck after cooldown.

Before investing in upgrades, execute the fundamentals—good results are achievable with stock hardware if you follow the sequences in Newtonian, SCT, and RC/CC sections and verify with the star test.

Collimation for Visual vs Imaging

The goal is the same—concentric, sharp on-axis stars—but the tolerances and priorities differ.

Visual priorities

• On-axis sharpness: Planetary and double star work demands on-axis perfection. Slight off-axis errors matter less visually.
• Quick touch-ups: A fast tweak on a bright star is usually enough. Eyepiece changes can slightly alter registration; compression rings help.

Imaging priorities

• Sensor orthogonality: The camera must be square to the optical axis. Use a tilt plate if corner stars are uneven after collimation.
• Backfocus precision: Particularly with correctors/flatteners (EdgeHD, RC flatteners, Newtonian coma correctors). Measure spacing carefully.
• Rigidity: Heavy cameras and filter wheels can sag. Threaded connections and robust focusers reduce flexure.
• Field uniformity: You care about corner stars. After collimating on-axis, diagnose corner differences; don’t over-tweak collimation to fix tilt.

Workflow tips for imagers

• Collimate with the full imaging train: Especially for fast systems and HyperStar setups.
• Use short exposures: Live-view or short video makes iterative collimation efficient.
• Quantify: Use FWHM/half-flux diameter metrics and corner star elongation maps to guide tiny adjustments.

Resist the temptation to fix every corner with collimation alone. Once on-axis is perfect, tackle mechanical tilt and spacing. Corner asymmetry that swaps sides with camera rotation is a tilt clue, not a collimation problem.

Common Mistakes and How to Avoid Them

• Chasing concentricity everywhere: In Newtonians, fast scopes require secondary offset. Perfect concentricity of all reflections is not the goal; correct geometry is.
• Using a miscollimated laser: Spin-test your laser in the focuser; the dot should not trace a circle. If it does, adjust the laser or use a Cheshire/Barlowed method.
• Skipping star verification: Tool collimation gets you close; the star test closes the loop.
• Overtightening screws: Can pinch optics or crack correctors. Make adjustments small and deliberate.
• Ignoring cooldown: Thermal plumes can mimic aberrations. Give the scope time, or use fans if available.
• Adjusting multiple things at once: Change a single parameter, recenter, and re-evaluate. Keep notes for repeatability.

Transport, Thermal Management, and Maintenance

Transport habits

• Support the optics: Transport with mirrors locked or secured per manufacturer guidance. Avoid vibrations and impacts.
• Recheck on arrival: A quick collimation touch-up at the observing site is routine, especially for truss or large-aperture instruments.

Thermal management

• Ventilation: Many Newtonians benefit from a rear fan to move boundary layers off the primary.
• Insulation or slow-cool strategies: In large temperature drops, insulating tubes or staged cooldown can reduce gradients.
• Dew prevention: For corrector plates, a dew shield helps maintain a visible star image during collimation and testing.

Routine checks

• Fasteners: Inspect spider vanes, cell bolts, and focuser mounting screws periodically.
• Markings: Ensure your Newtonian primary center spot is intact and visible. Replace if damaged.
• Clean optics sparingly: Dust rarely affects collimation significantly; clean only when necessary and with proper technique.

Good mechanical habits reduce how often you need to perform major adjustments and improve the stability of your Newtonian, SCT, or RC/CC collimation.

Field Checklist: Collimation in 5–10 Minutes

Use this quick routine before a session.

Newtonian quick routine

1. Insert the sight tube or laser; verify the laser hits the primary center.
2. Adjust secondary tilt/rotation if needed.
3. Use a Cheshire or Barlowed laser to center the primary return shadow.
4. On a star, verify concentric rings; tiny tweak of the primary if required.

SCT/EdgeHD quick routine

1. Pick a bright star near zenith; high magnification.
2. Slightly defocus; center the donut shadow with tiny secondary screw tweaks.
3. Near focus, ensure the first ring is even; finalize.

RC/CC quick routine

1. Verify laser returns on itself; adjust focuser tilt if present.
2. Align secondary to primary; refine with Cheshire if applicable.
3. On a star, tweak secondary/primary minimally; confirm corner star symmetry via short exposures.

If anything seems stubborn, consult the diagnostics in Star Testing and the mechanical advice in Mechanical Alignment and Upgrades.

FAQs

How often should I collimate my telescope?

It depends on design and handling. Solid-tube Newtonians often hold collimation well unless transported or temperature swings are large; check before each session and adjust only if needed. Truss Dobsonians typically require a quick touch-up each setup. SCTs/EdgeHDs can hold collimation for long periods if not jarred; verify occasionally with a star. RCs are more sensitive to small changes; check regularly, especially for imaging.

Can I collimate on a planet or the Moon?

You can get a rough sense of alignment with planetary detail, but the diffraction pattern around a star is more definitive. A star provides a point source with clear Airy disk and rings. Use a star for final collimation, even if you begin by centering and focusing on a planet.

Do I need a laser collimator for a Newtonian?

No. A Cheshire/sight tube and a center-spotted primary can achieve excellent collimation. A laser is convenient for aligning the secondary to the focuser axis, but it must be collimated itself. For the primary, the Barlowed laser technique or a Cheshire is preferred because it’s less sensitive to tool registration errors.

Is an artificial star as good as a real star?

For rough and moderate precision, an artificial star is very useful, especially when seeing is poor. For the highest precision, an actual star at astronomical distances eliminates finite-distance effects. Position an artificial star as far as practical (≥ 20× focal length), ensure it’s tiny, and elevate it to reduce ground-layer seeing.

What if my SCT’s collimation seems to drift during a session?

Focus shift and mirror flop can cause apparent drift as you change focus or target altitude. If your SCT has mirror locks, use them after focusing for imaging. Also, verify the diagonal and eyepiece/camera are securely and repeatably seated—registration issues can mimic drift.

Advanced FAQs

How precise must collimation be for a fast Newtonian with a coma corrector?

Very precise. At f/4, small angular errors translate to noticeable residual coma even on-axis if the primary axis is off. Use a high-quality autocollimator and/or Barlowed laser for primary alignment, ensure proper secondary offset, and include the coma corrector in your imaging train when verifying tilt and spacing. Finalize on a star at high power and confirm corner star shapes with short exposures.

Can I use a camera-based collimation aid instead of eyepieces?

Yes. Many imagers collimate live with their cameras using short exposures and ROI on a bright star. Software that measures FWHM and models star shapes across the field can guide adjustments. For SCT/RC systems where you don’t remove the camera often, this is efficient and reflects the actual configuration used for imaging.

I see astigmatism at focus—how do I tell if it’s optics or tilt?

Rotate components. If the astigmatism rotates with the camera or diagonal, it’s likely tilt or mechanical. If it remains fixed relative to the telescope, suspect residual miscollimation or optical stress. Check with the camera removed and a straight-through eyepiece to isolate variables, and ensure mirror clips/holders aren’t pinching.

Do I ever adjust primary tilt on an SCT?

No, not in normal use. SCTs are collimated via the secondary tilt only. The primary and corrector positioning/spacing are factory-set. If you suspect primary tilt or corrector misalignment due to damage, consult the manufacturer or a service center.

What’s the role of backfocus in EdgeHD/RC collimation outcomes?

Backfocus affects the correction delivered by internal lenses or external flatteners. Incorrect spacing can produce field curvature and off-axis aberrations that can be confused with collimation errors. Always set exact backfocus per the manufacturer, then collimate on-axis, and only then diagnose residual corner issues as tilt or spacing.

Conclusion and Next Steps

Collimation is the key to unlocking your telescope’s full potential. Whether you observe visually or capture deep-sky images, a systematic approach—proper tools, a consistent sequence, and a confirming star test—delivers crisp, high-contrast results. Start with daylight alignment for Newtonians, refine on a star; nudge SCT secondary screws with patience under steady skies; and for RC/CC systems, iterate gently while validating with short exposures and corner diagnostics.

With practice, the entire process becomes quick and routine. Keep a checklist, make small adjustments, and verify. For more hands-on skills, explore related topics such as mount alignment, focus optimization, and field tilt correction. If you found this guide useful, consider bookmarking it for your next session and sharing it with a fellow observer who’s ready to turn soft stars into pinpoints.